It's pretty clear, I think, that I'm about as ardent a First Amendment defender that you're gonna find. That includes the right to protest. If you're out in a public park, you're allowed to say any bullshit thing that you want to say and I'll defend your right to say it and I won't demand that you get permission.
There's an ethical problem with blocking roads in order to protest that doesn't apply to parks.
The worst thing that happens with an unannounced, peaceful protest in a park is that somebody will get offended.
An unexpected road closure like what happened in the UK last week can result in people not getting to the hospital in time. A woman is permanently paralyzed on the left side of her body now, not because protesters decided to block a street, but because they did it completely unannounced and people couldn't plan around it.
We close major streets for bike races and marathons all the time; but, they're generally well planned and announced so people can plan ahead.
If you think that your protest should be taking place in the middle of traffic, you should let people know as best you can that it's going to happen.