Parrots will only teleport to you when you're standing on a solid block (or possibly leaves, I forget if this was changed or not and when parrots were first introduced I believe they wouldn't teleport while you were on leaf blocks). Oceans are vast, however, so it's easily possible to go out of teleporting range by sailing across one.
That's possible, but the nether is a bit dangerous for that, well you could create a mine cart tunnel with bridges across the lava lakes but to create outposts for the nether portals for that purpose just seems like a royal pain in the backside. You do make a good point about the nether being a shortcut though.
If you go either up close to the bedrock ceiling of the Nether or down close to its bedrock floor, you can drive tunnels through nigh-uninterrupted netherrack for your minecart rails. This is easier and shorter than digging such tunnels in the overworld underground, although on the flip side you won't find any gold, iron, or diamond ore in the Nether. I generally recommend if you travel to or through the nether a lot, that you move your nether-side portals up to just below the ceiling of the nether and dig your rail or walking tunnels up there.
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Get the parrot to hop >into< the boat. Once a pet, (or other mob), is actually sitting in your boat they will not get out again until you break the boat. That has been the most reliable way to transport mobs long distances without inadvertently leaving them behind. I've found the best way to entice them into the boat is to get in the water yourself and swim around near the boat. Your parrot will follow and eventually settle in the boat.