You're sure you really want to do this? I stopped going to Parliament Hill on July 1 about 20 years ago. It is terribly crowded. If it is hot, people faint from heat exhaustion.
In fairness, there will be a lot of entertainment, but it will crowded and crazy.
If you want to go downtown, my advice is to get to one of the major bus stations early in the day. Suggest you check the OCTranspo website for details on bus transportation downtown. The best advice I can give is to park at one of the Park 'N Ride parking lots next to a bus station, but to select one that is the furthest away from the downtown core--possibly at Terry Fox (in the west) or even further out. All the buses fill up at the bus stations that are the farthest out, and then they just don't pick up anyone else at all until they hit downtown and start dropping passengers. It is pretty well the same deal leaving downtown--you may wait well over an hour to get a bus out of town.
If you have small children, I wouldn't do it unless you want to rent a hotel room downtown and stay there for the night. DH and I have done that, and it works out fairly well. You should still be able to get a reasonable room rate at a lot of downtown hotels for July 1 (don't try either the Chateau Laurier or the Westin--way too expensive, but a lot of other hotels in the city core aren't bad--there's a Novotel, a Hotel Indigo, a Delta and so on).