I'm not sure what the rule would be on a parent's legal name change AFTER the child was born, but I do know from some research I did a little over a year ago (when I was debating another member's claim of what he thought) that name changes (for non-marriage-related reasons, more on that below) done BEFORE the birth of said child do change how the name should be reflected on the child's BC (in most states).
The typical protocol is that name changes arising from a marriage (or other domestic partnership) should NOT be reflected with how the parent's name is entered (this is done because the name may change again later on due to divorce or remarriage, and if married names were entered both parents would usually have the same one and using the "maiden" name provides more identifying information). The rules are different if the name was changed for another reason (including gender changes) - here they usually DO want you to use the name after the legal change (a few states, including Oregon, want you to use the changed name only if it amended the birth certificate - contrary to what the aforementioned member thought it doesn't matter whether the old name is visible or not*, but just that the change was reflected on the parent's BC). *Removing the old name, if done, is for our personal benefit and does not affect the "validity" of the change for this purpose. The instructions can be a bit ambiguous for those with atypical name changes, and if you have any doubts you can ask someone at the vital records department (do not ask hospital staff as sometimes they aren't aware of the rules in special cases, as I rebuffed to the member I mentioned). If they ask for your "birth name" in this context that usually means the name as shown on your BC (accounting for any changes) and not necessarily the one given at your birth (which is the context usually used on these forums), although sometimes names changed for a reason that qualify for a BC amendment in that state (whether or not yours was actually changed) count as well (e.g. Wisconsin's policy). "Maiden name" means what your name would be without any marriages (so any changes to your first name would be accounted for). I noticed the majority of states phrase the question as something like "name prior to first marriage" or something similar - if you later changed your name for a non-marriage-related reason you probably ought to ask like I mentioned above about your case, but usually if you changed your first name that would be accounted for (I know a cisgender woman who changed her first name for personal preference and had to get this clarification).
