While you can't plant a forest inside a park once it is built (do it before! +1 appeal to other park tiles!)
You can plant Woods inside a National Park. Sadly, you cannot harvest anything inside a National Park, even a Marsh to put Wood instead.
your +20 tourism/turn national park doesn't look nearly as good.
Don't forget to mention that National Park isn't only for Tourism. It gives 2 Amenities for the city, and 1 Amenity for the closest 4 cities in your empire (basicly: like a luxury ressource). With the new Preserve district, the National Parks are even more valuable since you can now plan to put them around the Preserve and end up to have very good workable tiles.
In the example upthere, if the World Congress allows you to Culture Bomb with districts, I will put a Preserve on the Woods between the northern part of the Chocolate Hills and the Government Plaza of Amboise, steal the Rice tile (breathtaking) and the tile next to the city center of Chinguetti (at least charming).
Or you could conquer Chinguetti and put an additionnal National Park. The suzerain bonus of Chinguetti is incredible... if you have a religion. None of you have one, so you probably started the game at the Industrial era or later (so no Kilwa Kisiwani), and unable to enjoy the 6-Envoy bonus. That City-State has no real value, sadly, except of course for the 3 Faith per city with a Holy Site in a game without religion and third building, the +1 DF and some juicy trade route with Democracy.
But the best National Park improvement is the Eiffel Tower. If you have it, all tiles enjoy +2 Appeal, which means that almost all National Parks are +8 Tourism.