Dot there are 3 bridges over the river Ouse in central York, they are Lendal Bridge, Ouse Bridge, and Skeldergate Bridge. The only streets which your parents could have stayed in which border the river (I think) are North Street and Skeldergate. The 2 hotels I gave links to are in Skeldergate, the Queens Hotel is actually on the river front. Lady Anne Middletons is not. There is a hotel in North Street which fronts the river but from what I can gather it is not to be recommended.