I like to play with my parrots on my shoulders. Problem is, when I go fishing like that and cast the Fishing Rod, it grabs onto one of them.
The easiest workaround is to simply take the parrots down from your shoulders using the Fishing Rod. This isn't anything particularly game-breaking, however it can be annoying at times.

Platforms:
Nintendo Switch (9.1.0)
Android (9)
Windows 10 (unknown build)

More detailed description:
Expected outcome - Fishing Rod is cast and falls on the ground/in the water.
Actual outcome - Fishing Rod is on Parrot that is sitting on the player's shoulder.
Reproduction - Cast a Fishing Rod while at least one Parrot is on your shoulder.

Sidenote: No matter how many times I repeat this, the parrots don't seem to get hurt, while if I were to catch them using the Fishing Rod, under normal circumstances, they would lose 1 point of health.

I also included a video file among the screenshots, the one file without a thumbnail, file name 2019122610493301-11B64E28AD7A49CA9EC8AC007BE858C6.mp4.

Edit (31.12.2019.):

• Uploaded screenshots for mobile (Android)
• Updated platforms
• Happy new year (and deacade)!