Liseberg kind of has this type, but with entrance fee too.. Pay a smaller sum for the entrance, then you can enjoy the park without riding coasters.. Old people like go and dancing, there's dance hall. People go listen to their favorite bands at the outdoor arena/scene. Then to ride you buy a pass bracelet (most common) or buy tickets. Back in time there were tickets grouped A-E, only that.. Then tickets changed to number of tickets so you can use same ticket for all rides but give different amounts. I think it is still so now. But the bracelet is far most popular. There are lower cost for only the kiddy area, and normal cost for whole park.
Then there's special cards that you can buy for one whole season and ride everything. Popular with pre-teens and teenagers and ride fanatics. I used to buy but not anymore because now we use the handicapped "guide" for free when one nice cashier suggested it to us. So basically 2 for 1.
I think Gröna Lund in Stockholm is set up in similar way, and I think there's many European parks like that too. Different concepts.
Free entrance (no fencing) and pay to go I associate with mobile parks. And parks that is not really fleshed out amusement park, but rather a beautiful park with a tiny selection of rides.. like carousel.