Depends where you are - just looked at an institution known to me, let's put it that way, and the social sciences funding is indeed full-time only, the reason given being the intensity of the research projects. Hmm, I'm not going to comment further on that one here.
Just looked at the Research Councils out of interest and the ESRC offers part-time funding but over 5 years (not six as you might expect), but it is at least there. Is this a route open to you at your chosen institution? The ESRC also do collaborative doctoral scholarships, which would allow you to get some experience of a non-HEI if you are not wanting to go into academia, not sure if they are available part-time.
I agree that funding is discriminatory if you have a young family and want to do it part-time, add to that the structural inequalities of seminars and conferences being in the evenings or weekends and perhaps requiring travel and the expectation that you contribute to the research culture, and yes, let's just say postgraduate study does favour the mobile and commitment free, or at least those with domestic support.
Do you know any of the academics well where you want to study and do they rate you? What possibility is there of getting a PhD linked to a project grant, where there might be more flexiblity about how that is done, depending on the length of the grant of course?
As to the suggestion of doing it full-time, I would really only do this if you are prepared to and can commit the time. It depends on your institution, but universities are concerned about progress and completion because of things like the Research Excellence Framework. You will be expected to do a full-time study schedule, especially if you have a studentship, and complete within the required time.