Depending on how long you can use daddy daycare - I would be thinking about longer term care before you sort out hours - because if you have to use long time daycare for example you have to pay the set amount irrelevant of number of hours so it can work out expensive to use if you are working 5 short days, and the current capping of the rebate also makes 5 days not so great. This year I have one in school so working more than 3 days became more viable but prior to that the care costs with long day care in particular were prohibitive. Other forms of care probably work out better for shorter days. (you might already know all this, but I really was incredibly naive when I first went back to work (between DD&DS and then when DS turned one) and wish I had realized more of the implications of different jobs, different types of care etc.)

Also is the position a permanent, or fixed term permanent, temporary, contract etc.? because I think that makes a big difference as to how likely different options apart from full time will go down with employer. Contract which is what I have always done places have little to lose by taking reduced hours as are only paying you for when you are working - creating a part-time (so actual part time rather than flexible hours - so less than 38 working hours a week say) salaried position if you have approval for a full-time position in your budget is not always easy.

I assume you are actually wanting part-time as opposed to full-time with flexible working option? (as I said before, full-time flexible working (e.g. 4 long days, 5th day off, or short office hours + make up at home) from my experience is easier to obtain than part-time.