Still an issue. If you can get the parrot to dismount, you would be able to go through the portal, but you'd have to push the parrot through the portal when it's sitting, before you enter it yourself, if you don't want to leave it behind.
Tested on a bedrock dedicated server.
Effects the Bedrock dedicated server as well, when going between nether and the overworld. Reconnecting to the server fixes the issue though.
Still an issue. If you can get the parrot to dismount, you would be able to go through the portal, but you'd have to push the parrot through the portal when it's sitting, before you enter it yourself, if you don't want to leave it behind.
Tested on a bedrock dedicated server.