I honestly haven't picked apart the budget trying to figure out exactly how it's all cut up... but my guess would be that it is or it isn't "subsidized" depending on how you look at it.
Are elk tag dollars going for new seats on the toilets at Chatfield... no.
But I bet the departments share costs of various staff, buildings overhead, etc. There are probably certain accounts that are earmarked for certain types of spending and certain accounts that can be mixed.
I'm sure you could make a case either way, but I think the bottom-line is that it's way cheaper for the state if each were standalones, and because of PR elk tag money is more restricted in what it can be spent on.
Personally, I have more of an issue about the ridiculous amount of money spent on stocking non-native fish than I do on parks.