Yes, it is ok to pick in the state parks. My recommendation is go to the Missouri Conservation Departments website and try to find a conservation area close to where you will be. They are often smaller, but from what I can tell they are less traveled, but have no eveidence to back that up. You might find the smaller ones are closer to you than the state parks. Anyway, I found two today in a conservation area in Greene county (close to Springfield) today.
Here a link to the website, they actually have a short article on morels there too...not extremely informative just basic info.