I live close to the Indiana Dunes National Lakeshore, which is a federal park. I am able to take my dog with me as long as I have him on a leash at all times.
I'd suggest you call your state's Department of Recreation and ask them their policy, as it may vary from state to state. In most cases, I'd probably say that yes, they are allowed, but with some type of restrictions like certain areas, leash laws, etc. some dogs are not allowed in state parks bcause of the law.
Cape Henlopen State Park was created in 1964.
The area of False Cape State Park is 17.486 square kilometers.
The area of Cape Disappointment State Park is 7,616,183.7869568 square meters.
Bill Baggs Cape Florida State Park was created in 1967.
Point State Park was created in 1974.
Walnut Point State Park was created in 1968.
Colton Point State Park was created in 1936.
Cypremort Point State Park was created on 2004-11-20.
The area of Cypremort Point State Park is 748,668.438144 square meters.
The area of Tawas Point State Park is 740,574.7252992 square meters.
Jones-Confluence Point State Park was created in 2001.
The area of North Point State Park is 5,341,850.477568001 square meters.