Alright, so I redid everything in vanilla singleplayer. Luckily, the only issue was the one that is working as intended. However, this could be improved upon. Jumping down is essential to play MC, and it's annoying to have parrots fly off after jumping down just one block. It would be a lot nicer for players to have a hotkey to pick up parrots and put them down, instead of having them automatically get on and off, just as SHIFT is required to ride horses.
Also, I don't know how a modified server that only affects the player's position would make the parrots stop sitting and start teleporting. That should probably be looked into, despite not being vanilla, since it's on almost all survival servers.

The third issue sure works weird, then. It doesn't make sense that the birds would just fly off while on a player's shoulder. Also, I've encountered another issue in _vanilla singleplayer _(still 1.12.2); if flying or sprinting quickly, the parrots die/suffocate in surrounding blocks when they teleport to the player's location. I will test everything else again in vanilla singleplayer to see if the issues still arise.

Your third issue is working as intended. Additionally, you are playing on a modified server. We only accept games for the vanilla game here.

I stated three issues, but there were only two duplicates, both of which have not actually been resolved. Parrots also still dismount when hitting downwards or diagonally downwards (this is the worst issue to me, as they're hard to find, and it's extremely easy to accidentally kill them when they fly straight into where you're hitting).