Why does the LDS church seem so against a rape victim (or other unplanned pregnant woman) becoming a single parent?

This is a delicate topic which must be clarified. There are myriad ways for a woman to become a single mom. Some hold more stigma than others.

Rape victims have done nothing wrong. This is one circumstance in which the Church condones abortion. If the victim decides to carry to term, they are urged to place the baby up for adoption so it can have both a mother and a father, and be raised without knowing of the rape. Rape victims do not need to go through any special repentance process before they are considered worthy members of the Church. If the victim places the baby up for adoption, she may even serve a mission.

Some women have children with their husbands, then there's a divorce, and the women find themselves single parents. The children stay with a parent (possibly joint custody). These single mothers need not repent for being single mothers, but may need to repent for what led up to the divorce (if it was their fault).

Finally we have single women who are sexually active and become pregnant. It doesn't matter if the pregnancy was planned or not; breaking the Law of Chastity requires repentance. These women are strongly encouraged to carry to term then place the baby up for adoption where it will have both a mother and a father. Also, the woman has better opportunities for education without lugging a baby around.

No one questions the mother's ability to love the baby, but the Church wants what's best for the mother and the baby.

Mini Kilt, on the Single Parent issue are you referring to a specific incident or series of incidents? I know of the informal social stigma of single parenthood but officially the LdS Church seems aligned with other religious groups in this regard. . .

" Because of a variety of circumstances, including death, divorce, and separation, many Church members find themselves in single-parent families. Regardless of their family situation, all Church members are entitled to receive all the blessings of the gospel of Jesus Christ. The structure and fellowship of the Church provide many people who are willing and ready to help parents and children and to strengthen them in the gospel.

Addressing single parents, President Hinckley said:

"This is a lonely duty. But you need not be entirely alone. There are many, ever so many in this Church who would reach out to you with sensitivity and understanding. They do not wish to intrude where they are not wanted. But their interest is genuine and sincere, and they bless their own lives as they bless your lives and those of your children. Welcome their help. They need to give it for their own sakes as well as for your sake.

"Single-Parent Families

The First Presidency and the Quorum of the Twelve Apostles have taught that "marriage between a man and a woman is ordained of God and that the family is central to the Creator's plan for the eternal destiny of His children" ("The Family: A Proclamation to the World,"Ensign, Nov. 1995, 102). While this family unit is the ideal, many Church members find themselves in a variety of other circumstances, including in single-parent families. The gospel of Jesus Christ was given by God to bless all His children, without exception, regardless of the family situations in which they find themselves.

We have thousands of good bishops in this Church. We have thousands of good quorum officers. We have thousands of wonderful Relief Society women. We have home teachers and visiting teachers. They are your friends, put in place by the Lord to give of their strength to help you. And never forget that the Lord Himself is a source of strength greater than any other" ("To Single Adults,"Ensign, June 1989, 74)."
http://www.lds.org/ldsorg/v/index.jsp?locale=0&sourceId=b30df73c28d98010VgnVCM1000004d82620a____&vgnextoid=bbd508f54922d010VgnVCM1000004d82620aRCRD

. . . I can't find any "official" information on single parenting as it relates to rape but I'll keep looking.

I would think that's a tough situation for anyone but given the tight "small town" nature of Mormon Culture I can see how the social stigmata of becoming a single parent as a result of a rape could be perceived as "oppressive" by some. However, I'm not sure that Mormon Culture is all that different than mainstream culture in that regard.

BTW, great question. +6

SHORT ANSWER:
Because a rape victim who chooses to become a single parent grates against the "worthiness" standards of private Mormon Culture regardless of what public official sources may say.

LONG ANSWER:
Thank you for your clarifying comments on my first answer Mini Kilt. I know what you're talking about now.
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Basically what you've described is what falls under this category in the BITE model for determining if a group is a Mind Control Cult or not:

Section I.
Behavior Control

Criteria 5.
Rewards and punishments (behavior modification techniques- positive and negative).

As ExMormon and Exit Cult Counselor Luna Fletcher notes:

"Social pressures can be very strong. This topic is a mixed bag, as some members are sincere, good-hearted people, and others are judgmental and shaming. The overall vibe, however, is to appear righteous, or you are unworthy, inferior. It is a part of the culture.

Members and families who seem to be following the commandments very closely are highly respected. Those who have visible problems are considered inferior. This is never expressly said, for the doctrine is "Love one another", and we're all equal in the sight of God, nevertheless various levels of status exist in the culture very strongly.

One could be considered of lowered status if: a child in the family is rebellious or has left the Church, one of the parents is inactive or a non-member, any member of the family is observed visibly disobeying a commandment (drinking/smoking, dressing immodestly, criticizing leadership, not wearing garments, turning down a calling, cussing, missing a lot of meetings, etc), if the family is poor (though not always), if no one in the family has ever held a leadership calling (President or Councilor of an adult group, Bishop, or higher), or even if they just don't seem "valiant".

The threat of social ostracism is implied by how people talk about others. The language is loaded with negative words for people who sin, criticize, or leave. "Apostate", "wicked", "sinner", "blind", "hard-hearted", "unrepentant", "deceived", "mocking", "rebellious", "contentious", "misguided", and "spiritually weak" are just a few.

Some phrases use guilt to punish, i.e. "By sinning, we crucify the Savior anew".

Spiritual/intangible rewards and punishments abound. These include blessings (financial, situational, spiritual, physical, etc), The Spirit (to enlighten), promise of eternal heavenly life, and promise to be with your family forever. Spiritual punishments consist of the loss of these things. Satan is targeting members of the Church, so you are at risk of being tempted or even harassed by him. Mormons have a very real fear of evil spirits, and are inclined to see the hand of Satan in any tragic circumstance."
http://www.concernedchristians.com/index.php?option=com_fireboard&Itemid=42&func=view&id=80003&catid=532

I think that what you've observed is merely an extension of the behavioral patterns that Ms. Fletcher documents in her article.

Another ExMormon and Social Scientist, Blair Watson has noted that for a woman there is only ONE model that is deemed "worthy":

"LDS women are depressed is that they have been psychologically conditioned by Mormonism to base much of their self-esteem and identity on being a daughter of a male deity {'Heavenly Father'} and a 'wife and mother in Zion'. As mentioned, Heavenly Father is a psychological construct, the product of human thought. To base part {or all} of one's self-esteem on something that is only a belief is not wise. It is also not prudent to base one's self-esteem on one's marital status, which many women, in and outside of the LDS Church, do. Why? Well, what happens if your spouse is killed? With your husband gone, is your self-esteem going to collapse? What would happen to your self-esteem if you discover that your spouse has been having an affair? Will it crumble? Furthermore, if you base your self-esteem on being a mother, what happens to it if you're no longer a mother because your child is hit by a car and killed? These aren't pleasant things to think about, but the truth is that they are realities for some women, including some LDS women, and heart-breaking situations happen to people. Life is full of risk."
http://www.concernedchristians.com/index.php?option=com_fireboard&Itemid=42&func=view&id=74805&catid=526

Since being a single parent falls short it can generate scorn resulting in a loss of social esteem, shame, even implied guilt for the "unworthy" woman. And THIS will never do!

So as misguided as the shrill, judgmental voices might be believe it or not from the Mormon System they actually think that they're doing your friend a favor by encouraging her to put the child up for adoption. But doing so she gets a kind of "social reboot" and at least returns her to "ground zero" so she can start building up "worthiness chips" again.

So in the end what we end up with is a major incongruity between the "public" face of Mormonism - as represented in the clip from the www.lds.org website in my prior answer - and the "private" face - as represented by the analysis by Ms. Fletcher and Mr. Watson above.

While some Mormons recognize this incongruity - some have openly criticized it - others are utterly blind to it. I hope that your friend finds and gains support from some of the former and avoids the latter.

SUMMARY CONCLUSION:
To keep a child that is a result of a rape is to willing choose to live in and with your "unworthiness" - which is Mormon Culture is generally considered poor, even unwise judgment.