For our family of 5 with a 9, 6, and 2 yo....we budget....
$35-$40 for a generous counter service meal (3 kids meals and 2 complete adult meals if we would like it.) Add $10 for extra "adult meal" or $6 if it is a "kids meal".
Snacks: Well that depends...$6-$12 for 1 or 2 popcorns, or $20 if everyone is getting ice cream from the cart ($3-$4 or $5 per item)---per snack trip. So another $30-$40 on snacks.
Water: It is expesnive. We freeze bottles and put them in the bottem of the stroller. We have run out of our bottles (we packed a good amount!) the past couple of times, so end the day with several $3 bottles of water.
For a day visit, we plan on $100 to cover 1 meal, snacks and any surprises per day. This includes spare water. If including your water--I would add another $20-$30 just for water purchases if you plan on buying it. Water is essential, so have a plan if you wish to save money here.
Kids have their own spending money--we pre-allocate our contribution to souvenirs if it is a vacation. Usually $20-$30 per child if a "real" vacation. And at minimum, they get nothing if it is just a day visit.
Parking--I want to say that was something like $15 when I read the sign last time we went.

You pay once per day you visit.
We usually carry a debit card and some minimal cash for minimal things. We carry cash for the kids so that they know once the money is gone, it is gone. Makes it easier to not cave for the extra $1 or $10 for a coveted item.
I pre-plan our mom and dad spending keeping in mind what we would like to purchase ahead of time. Which lately, hasn't been much.
So I would say--$150 per day at the parks is "safe" not including souvenir money.
For a vacation, I wouldn't do less than $20 per child which safely gives them a good selection of a toy, silly hat, a shirt, an animal or craft kit. We do not let them buy souvenirs first thing and will always hit the big store on the way out. If however several stores they hit upon a unique item in a specialty shop, we will let them get it. But we have a no return policy. (Disney will let you return as we have discovered, but now the child has to make sure they want something and understand that we will NOT return it.)
Other options are to pre-buy Disney souvenirs at
walmart so that you can get more, but my children (as do I) love a souvenir from the actual place. But we try to shop prudently to avoid having a piece of crap trinket when we get home. They have learned to be very particular about items.
ETA: And this is me not being frugal. We have gone to Disney for the day for less buy packing a soft cooler with sandwiches and snacks. I have gone to the parks without paying a dime before!
Other extras include carnival games at AK in dino land and video arcade games in Tomorrowland and the shooting arcade in frontier land. I do not recall the costs on these, but you can always have the kids use souvenir money for this purpose.