I live in York itself, but that doesn't mean I don't understand some of the options for visiting the place. My own house is unsuitable for putting up friends or family, and so we're always going "half-tourist" when they want to come and stay.
I'd say the first consideration is what mix of activities you want to undertake on your holiday. If they're all, or are predominantly, based around the city centre (guided tours, walking the walls, museums, galleries) then I would strongly recommend looking close to the centre. This need not be within the walls themselves (I can think of only a few hotels that are) but there are a number of hotels or guest houses within short walking distance of the centre and most will offer some level of parking. If you want to be in the centre most days, just take it easy and cut out the hour or so of driving/bus travelling1.
If your plans are more wide ranging, with visits to locations outside of York's outer ring road (say, visits to Scarborough or other coastal locations, trips to the moors or to Ripon) then I'd recommend looking for villages outside the outer ring-road but close to it. (For instance, my parents once rented a cottage in Stockton-on-the-Forest and I actually moved out of my house and went to stay with them for that week).
Here I'd draw your attention to York's outer ring-road. The South and East sides are provided by the A64 dual carriageway. The North and West are provided by the A1237 which is single carriageway. A lot of more recent housing developments have been happening on the West side of York and it's generally acknowledged that the A1237 should be dualled2 at some point in time but isn't currently. The West side can also become clogged if there are certain events leading to (partial) closures of the A1(M) (of course, we cannot predict whether these would happen during your stay)
That means I would focus me attentions towards the South and East, unless you're planning a lot of excursions to the West. You might consider being on the North side if you plan multiple excursions into the moors.
If you're staying towards the South or East, then at least 3 (4 if you count Monk's Cross) of the Park & Rides are easily accessed from the A64 and which one you choose to use will not make a great deal of difference. I would note that when I did this with my parents, the cost of 3 adult tickets for the Park & Ride was actually slightly more than we would have paid for city centre parking. This was some time ago and the situation may have improved. Also bear in mind that they do offer family tickets and those should be more economical if you qualify.
1Also, if you do end up walking a lot around the centre and have small children or people prone to getting tired, don't underestimate the value of being able to pop back to the hotel/guest house and have an hour's quiet nap.
2Upgrades to York's outer ring road:
Dualling of the York Outer Ring Road is something we hope to carry out in the future, however we don't have the level of funding to do this at this time.
This scheme is a “stepping stone” to future dualling of the YORR ...