Suggestion for accommodation in York

It will be very difficult to fit all that in in a weekend. At the bare minimum it's three full days.

there are places to stay in York with parking, but they will need seeking out - many don't.

They do have long term public parking near York train station. We stayed once in a townhouse B&B outside the Wall. Check out the smaller neighboring towns for better deals on B & Bs with Parking and do York as a Day Trip. Some of York has no driving zones.

We stayed in Hull (officially Kingston-upon-Hull) because we wanted on-premises parking and less expensive lodging (York lodging is quite expensive). It's a 1-hour ride to York on the train. We stayed at the Park Hotel and Apartments, which was excellent and very close to the train station.

The Minster Hotel has parking. https://www.yorkminsterhotel.co.uk/

The Milner York, which is the hotel attached to the train station also has parking:

https://www.themilneryork.com/

I highly recommend the Abbey Guest House in York. We stayed there in June, 2023. It's across the bridge from the train station and, if memory serves me, there is parking nearby.

Last fall we stayed in Number 34 York with street parking, and this spring we stayed at Ashfield House in Grassington, with on-site parking. We would stay at either one, again.

Thank you everyone.
We have booked the Rose and Crown in York. It does have on site parking.
We also decided to incorporate Grassington into our next drive to the Lake District, after York.
Thx

We stayed at the Minster Hotel that has parking and it was nice and not expensive.