If your parents are into history, there is loads to do around York.
The Minster is always lovely, if a bit expensive. If you do go, I would recommend going down into the crypt because there is the many layers of York down there. It has Roman, Viking, and a bit of Saxon stuff down in there. Lots of fun and amazing stuff.
The Jorvick is super cheesy but I love it with it's cheesiness. However, Dig is pretty good too and has pulled up some Roman Gladiator finds.
You can walk the walls, which is a gorgeous way to view the city, even if you only do a little bit. There are also so guided tours around the city which can be hit or miss and there are some fun ghost walks around.
The Merchant Adventurer Hall is lovely, very gorgeous Tudor building full of history. It housed a Medieval hospital which has loads of interesting stories going in there, if you get a good staff member who knows a lot.
The Castle Museum has history of all ages. It has a Victorian Street in there, plus the prison area where Dick Turpin, famous highway man was held. Very fun, but takes a long time to get through.
The Shambles is beautiful and if Betty's is busy, there's a fab tea rooms there, including one I like a bit better than Betty's, especially if you want high tea because you get more food that's locally sourced and yummy.
There's the shortest street with the longest name Whip-ma-whop-ma-gate. It's kind of a fun one to see, and just off the Shambles.
Fairfax House is usually dressed beautifully and is a touch of the 18th Century town house.
There's the river walk, which is a lovely place to walk and you can get some ice cream.
If your parents are like mine, where they like to look a little and relax a little, there are tons of characterful pubs around with lots of history. A few popular ones with the tourists like Lendal Cellars which was the wine cellars of the monks.
There's the museum gardens which has the ruins of St. Mary's Abbey which is another gorgeous place to walk and look around. It's beautiful when it's sunny and lots of people take their sandwiches out there on sunny days for lunch. And behind Museum Gardens is King's Manor which is now owned by the University of York, but is open to the public. The council of the North, which was the King's right hand in the north, met there after the Pilgrimage of Grace until it was dissolved a few centuries later. It's a beautiful Medieval civic building.
If you have a car, you can always get to Castle Howard which is popular because Brideshead Revisited was filmed there. It's lovely and the grounds are beautiful. I also like Beningbrough Hall, which is closer to York and you can get there by bus, but it's a pain. Beningbrough has costumed actors walking around talking to visitors and it doesn't have as many visitors, but it has lots of paintings on loan from the National Portrait Gallery. Up near Ripon is Fountain's Abbey which is gorgeous. I love the water gardens. Harewood House is near Leeds and is also a lovely stately home, but with paths for walking. It also has a racey past as the family that built it made their money from sugar planatations in the Caribbean.
Anyway, that's my suggestions. We love the heritage sites so it's really heavy on those, but we're History MAs, so we really enjoy that sort of thing.