Booking anywhere in Italy during the summer is nearly impossible, but May is perfect; Venice is getting more difficult to visit on the fly now that travel has picked up again, although, so far they are a little below usual accomodations. Unfortunately, this usually means the high-priced places are the ones with the vacancies.
Also, if you're going with your folks, they might enjoy knowing where they are staying and for how much in advance.
In Venice, one of my favorite spots is a little out of the way--La Calcina is on the Fondamenta zattare Gesuati. They serve breakfast in the garden in warm weather, and it's one of the spots that Ruskin stayed. They also have a rooftop terrace, although, once you get there, you have to reserve a spot for sunset (no charge) so they don't have a huge crowd. They have nice rooms, several with terraces, and generally larger than many.
Ph-041-5206466 and this is the website for it--
http://www.lacalcina.com/HTML/en/calcina_menu_en.html
I also like the Albergo Doni if you want to be in the thick of it. Doni is a few canals away from the Piazza, and very close to the amazing seafood trattoria La Rivetta. It's on a side canal that many gondoliers use to access the Grand Canal, so sometimes you'll hear the singers as they go by. Simple and quaint. You can have a courtyard or canal view.
Best to call in their early afternoon or late morning to get an English speaker. You can also fax at late night to the same number. Ph/fax - 041-5224267. You can book and find info at this site
http://www.veneziasi.it/ita/hotel.asp?ID=DONI or you can just google albergo doni venice and it will lead you to some reviews and sites.
I tend to stay in Dorsaduro or Castello-in the Venetian action rather than tourist one. In Dorsaduro, if you can stay in the main area and not the annex, I rec. Antico Capon. Wonderful people and right on the Campo Santa Margherita, which I adore. 041-5285292
I also know of a great small apt. perfect for 3, but I think I only have the info at home, so it will have to wait. I've got a few rec.s for Bologna and nothing on Parma, but you can always trust Osso on food and place!
I'll try to get back here tonight to give you more info.