I hear and read these three words used in the same context in English, but is one of them more correct than the others?
Possibly there are dialectal contexts for picked and perked, but so far as I'm concerned they look like mistakes made by people who simply misheard or misremembered the original and correct pricked up.
Feasibly someone might pick up if they've been poorly, though I would normally expect that to be perk up anyway. Certainly in general to perk up just means to become more lively, and you can't apply it as a verb to your ears.
From OED: prick noun - a sharp point or spur, from which among various other meanings of the verb form, to prick up - to rise or stand erect with the point directed upward; to point or stick up. This usage has been around since at least 1775 - [the horse] pricked up his ears.
Here's an NGram showing that only pricked up is really used.