There is park land there and there is huntable land there. I live close and have hunted Taylor since the mid 80s.
You can look it up on the county web site
http://www5.kingcounty.gov/parcelviewer/viewer/kingcounty/viewer.asp and you can start with the parcel viewer here.
You can identify all owners of the parcels. There is mixed ownership all over the mountain. The park area is inside the gate where Iss-Hobart Rd intersect. You don't want to hunt there.
Other areas up the mountain are not park land but they are mixed ownership and not all owners want hunting on their lands. The private owners of the other parcels are mostly corporations and LLCs. Companies like Fruit Growers Assn has been neutral in the past and they are based in CA.
The land changes hands on occasion so it is up to us to do our research.
If you call Fish and Wildlife or the local Sheriff, you will get mixed info from them. They don't all know the rules and laws of that area (and many areas for that matter). You will get the run-around and get disgusted with improper info.
The signs on the gate tell it very well - it is regulated by our hunting regulations. I would start with the regs and identify where you can use a gun and where the rifle restrictions are.
Happy hunting!