I cannot say I have met anyone who supports abstinence who actually support limiting sex education to "don't do it."(as seems to be the common perception). Now they may not want the SCHOOLS to be doing what they believe is a parents' job, but the ones I know with that belief still do not limit themselves to "don't do it" when educating their children. There are probably those out there, but I am quite certain they are a small minority.
Most who support abstinence education support it as abstinence preferred education, which include teaching about the biological facts of sex, the social aspects of early sexual activity, the potential social and psychological consequences of early sexual activity, the medical consequences of sexual activity, use of birth control methods and their effectiveness in preventing pregnancy and STDs, (like the fact herpes, HPV and other viral STDs are not stopped by condoms, or any other birth control method) how self esteem (or lack of it) can play into the decision to have sex, etc, etc, etc.
Such courses give students everything they need as far as knowledge of birth control, etc. but add a WHOLE lot more focussed on the decision whether to become sexually active or not, as well as the innumerable consequences that can and do occur in addition to STDs and pregnancy.
OTOH, sex ed that excludes abstinence usually is limited to biological facts, warnings on pregnancy and STDs, and advisory one the type and use of birth control, and in some cases, "class is ended, have some rubbers."
LOTS of facts are left out in current traditional sex education classes, especially the social and psychological ramifications, many of which take a teen by surprise with negative consequences even if they do avoid STDs and/or pregnancy.