
This one isn't a playground, but it's a kid activity worth knowing about. The Marionette Theatre lives inside a very cool old building, the Swedish Cottage: a model schoolhouse built for the 1876 World's Fair in Philadelphia that has been put to all sorts of uses in Central Park over the years. It just got a new cedar shingle roof, which smelled great. Unfortunately, the outside was covered in scaffolding, so you'll have to take my word on how charming it is.
We saw Little Red's Hood, an updated Little Red Riding Hood. Red's a city kid addicted to her phone, Grandma's a spunky visitor to NYC, and the wolf is some kind of escaped lab animal who loves cupcakes. Familiar, but with a twist – like a cupcake. Or Toy Story 5.

It's a real, live marionette show, which is pretty cool. You can see the strings, and the theater is small and old-fashioned in the best way. The show runs about half an hour, and at the end the puppeteers do a little demonstration of how the marionettes work. The one strange part is that all the lines and songs are pre-recorded, so there's a disconnect between the live movements and the canned soundtrack.
Most kids in the audience were between 3 and 7, and kids under 2 get in free. My 3½-year-old enjoyed it a lot. Out of about 30 kids, 3 cried and had to be taken outside, which seems like a pretty normal rate for a dark room and a wolf.
Apart from the pre-recorded audio, here's my one real complaint: it started almost 15 minutes late. I showed up 15 minutes early expecting a prompt start, so we sat around twiddling thumbs for a very long time. At least we got the best seats.
There's stroller parking inside the building, and there are bathrooms, so logistics are easy.

Verdict: a sweet, old-fashioned outing that's a nice break from the playground. Just bring a snack for the wait.
On the run: that's how we came. It's just off the main Central Park loop on the West Side, near 79th St. By bike: same deal; the loop drive gets you right there. By train: B or C to 81 St–Museum of Natural History, then walk into the park.