Does it cost to park there?

Yes, about $10, not cheap for St. Louis, although maybe cheap compared to Chicago or NYC. You can park free in Forest Park, then go through the Planetarium to the bridge over the highway to reach the Science Center.

Paid parking only. Street parking is limited so chances of parking there will be tough

There is a small free lot on the planetarium side

There is a paid lot and limited on street parking.