Not really. Anti-abortion folks have tried to circumvent Roe v. Wade several ways--"informed consent" (read "having a non-medical person talk the girl out of the abortion"), requiring the father to consent (that was rejected by the courts also), notification of parents of minors (an open question, but the courts so far have said "no").
If I recall correctly (i follow crim law, not generic consitutional) in Roe v. Wade, the court determined that the state of the art (circa 1973) was that abortions were safe for the first 3 months, not that bad for the next 3, and possibly real dangerous for mommy the last 3. Similarly, the foetus was not viable the first trimester, probably not the second, and quite likely was the final trimester. The court found that the state had NO interest in regulating abortion the first trimester, some for the second trimester, and a strong interest for the third (as the viable foetus might, I guess) be seen as potentially a separate person with attached basic rights.
I'm not sure what the South Dakota law entails nor, as a current resident of California do I care but--all the above being the state of the law, the original decision in Roe v. Wade was a 5-4 vote in the Supreme Court. While all lower courts (state and federal) have to apply the law, perhaps the opponents actually think that the two new appointees will vote to overturn a 32-33 year old precedent. I'm not so sure.