Completely off topic from the main thread but would you be comfortable sharing how you found and/or convinced an advisor to take you on for a part-time PhD?
I spent a not-insignificant amount of time emailing professors (roughly two professors per state) across the US asking if they had part-time PhDs. Most didn't respond to me, the ones that did gave me a polite "no".
I then look up "online part time PhDs" and found basically two places that offer a doctorate online, which was Purdue and University of York (England). I inquired about Purdue, and it seemed more managerial than technical, which I wasn't interested in.
I find a program that seems interesting at University of York and ask them if they have any openings for a remote part time PhD. They responded back with basically "well not if you want funding." I then replied with "What if I fund myself?" Two professors working together decided to give me a shot, so they asked me to write a research proposal, which I did. I guess it was good enough, because when I emailed them back, instead of saying "thanks but no thanks", they gave me tips on how to refine it. After a few months of back-and-forth working on the proposal (and finishing my undergrad), we got the proposal into a state we were happy with.
After that, I just applied on their online portal and managed to get an intake interview. The intake interview was basically "<x> is involved with this PhD. Are you OK with that?" or "What if you encounter <y>?" I guess I did well enough because I got an offer the next week. I start in April.
----
Sorry for the life story, but I figured you might be able to be able to save yourself a few steps.
Not parent, but part-time PhDs aren't entirely uncommon (at least in the UK, where I am) and I'm currently supervising one.
There are two major difficulties and one major advantage over the full-time version. The advantage is that you can work, so students aren't dependent on external funding or an ever-shortening runway of their own funds. This isn't a small thing. On the other side, though, it doubles the length of an already lengthy process (do people really want to be spending 7-10 years of all of their spare time unpaid and lonely?) and increases the chance the the candidate will drop out (in that they keep an alternative on-hand, rather than burning their employment bridges to do it full-time). Given how hard the PhD process can be, simply saying 'sod that' is always a temptation.
Anyway, it'll depend on the supervisor and the university. Some have policies against it, others are happy to consider.
