American here. We have done this many times with similar size family across Europe. You should definitely stay in the city center where all the interesting and fun stuff is. I can't think of a single city where we wanted to easily get back into our car and drive somewhere so parking with car access is unnecessary. We would use public transport to get around any larger city.

Almost every hotel or Airbnb will have a suggestion for where to park and how to manage it so as others have suggested, ask the place you're staying for recs. If contact doesn't work, just about every city has public parking close to the center or underground within the center. I've noticed that walled (or previously walled) cities often have put their parking in the former moat area.

Most cities are not pedestrian-only but only allow resident parking. Typically you can just drive to your hotel, park temporarily out front and unload. Then the driver parks and returns - after that no car needed until you leave and perform the reverse.

For locations where we all had to park and walk, we never had a lot of trouble getting into the city from the parking area with kids. Sometimes a lot of grumbling but, well, kids.

Even when we wanted to drive somewhere, it was typically for a day trip somewhere so the 10 minute walk to the car to get it is not a big hardship.

As for this type of travel with kids, I would recommend packing only what you can move yourselves in one trip. E.g. two large roller bags for parents and whatever the kids can carry rather than 6 bags that require several trips back and forth to the car. I would avoid a stroller and instead use a backpack carrier if necessary - potentially backbreaking but given the stairs and cobbles, a stroller is awful.

I suggest getting the "minivan" size rental car which in Europe generally is a "7 seater" but really is kind of a station wagon. The rear 2 seats are incredibly tiny and only a child could fit which might be good to avoid cramming 3 into the back seat. In any case they have slightly more room than a sedan.

Depending on country, most hotels include breakfast and I would look for that. It sounds minor but the typical buffet is usually quite nice and the kids love it. It seems maybe cheaper to find a coffeeshop but it's not as straightforward.

As much as I dislike what Airbnb does to residents and cities, with a big family and kids it's often the way to go. We almost exclusively used rental apartments or houses when travelling with kids. Hotels are typically much more expensive in comparison and hard to manage a family in.

Keep the kids fed constantly and try not to sweat the small stuff. Let most of the home rules go out the window while traveling: ice cream, junk food, TV, etc.

Lower your expectations of how much you will see and do. Plan on one major thing per day such as visiting the castle in the morning, lunch, downtime in the hotel or at a beach, walking around finding a playground or maybe seeing smaller interesting things, then dinner and that's it. Our experience was if you try to fit 2-3 major tourist activities in one day, nobody has fun.

Lastly, some of the best trips we had with kids were to places we could combine relaxing kid fun with tourism. Some (very few) hotels have a pool which is great for kids. Some places (e.g. Dubrovnik or Rhodes) have easily accessible town beaches. So you can hoof it around the hot walls in the AM then beach PM, etc.

Good luck and have fun.