It'd probably depend on the park and how it's designed/managed, but I'd be shocked if monocultured lawns sequestered any carbon. I know in agriculture it's a huge problem that industrially-grown monocultures -- where they till the soil and crop-dust fertilizers and pesticides and herbicides and fungicides -- emit huge amounts of previously-sequestered soil carbon. A result is that doing the reverse -- i.e., growing food regeneratively in polycultures and without tillage and artificial fertilizers and without all the -icides -- is considered a good way to sequester carbon.

Considering we grow grass lawns similarly to how we grow corn monocultures, I'd bet grass lawns are similarly awful for the soil and thus the climate as well.