Yes, the solution is to label everything. You should label your comment as known to the State of California to cause cancer, or birth defects or other reproductive harm.
If the whole point of your account is to make people laugh with the imaginary pretense that your posts are coming from another person/institution then yes, the potential readers should know it's fake.
A comic strip artist that I follow constantly puts fake words into famous peoples' mouths in his newspaper sections, but even tough I know it's fake I can still partake in the momentary delusion and get a good laugh
If potential readers cannot differentiate between reality and parody they're either deep in a conspiracy rabbit hole, or reality has become too crazy. Both these cases would again be a valid reason for parody to point out this fact.
Jokes work by surprising the audience. Pretending to be someone else before revealing the truth is just one method of doing this, and requiring to label it as parody is like explaining the punchline before the joke. The Onion recently wrote an actual legal document on exactly this and how required labelling of parody violates free speech.