Parking in "the urban core" is usually not set up with tow vehicles in mind. Even delivery vehicles often have a hard time reaching their destination, so you have to think a little differently.
As pointed out above, box centres usually have far more parking than they need, however, these tend to be away from the areas that are most desirable to actually see and spend some time at.
Another idea is to make better use of commuter lots along the highways.
Usually a park and ride or commuter lot is serviced by public transit. The goal of the commuter lot is to get people from downtown and the 'burbs onto the highway without having more cars on the local roads. At night the commuter lots are mostly empty, through the day they fill, but usually they are designed with extra capacity.
Park in the lot (admittedly very far from downtown) catch a bus to the city centre and enjoy your day.
The advantage of this over parking at a box store, is that transit doesn't tend to serve box stores very well. So you find the parking, but can easily be stranded in suburbia. Of course if you carry bikes or don't mind a little walking that may not be an issue, and conditions vary from box-store to box-store and commuter lot to commuter lot.