If you're looking at staying 2 nights you could make the most of it by visiting different areas on the way there and back which would depend on where you're coming from.
But for example, set off early in the morning on day 1 and go to York on the way there.
Arrive at accommodation, dump bags and go out for a wander, arcades etc, perhaps beach , pick up any supplies, dinner etc.
Day 2, more time in local area and perhaps a short trip out eg water park at Scarborough or Bempton Cliffs RSPB reserve just north of Flamborough with thousands of birds nesting on the cliffs.
Day 3, pack up and morning in the local area before departing, visiting eg Whitby then drive home over the North York Moors, you could drop into the villages of Goathland and/or Grosmont and see the steam trains or even have a ride on one.
Or if you're happy to go a bit further north you could also take in Saltburn which has a pier, and tram up the cliff.
Danes Dyke, a steep wooded area that leads down to the beach is a nice place to visit just south of Flamborough or further north is Ravenscar, which is interesting as in the late 19th Century they laid out roads for a new resort that was never built and you can walk around, read information and walk on the old railway path.
But all these places get very busy on any day when the weather is good, which has quite a low bar in Yorkshire, so if you're looking for parking you probably want to aim to get there by mid morning at the latest.
Some caravan parks have on site pools and splash parks but you might struggle to find a 2 night booking at short notice in peak season.
A private room in a Youth Hostel could be a good budget option, there's the Boggle Hole one, but I don't know of any others in the area but obviously look on the YHA website. You'd also have access to a kitchen to make picnics.