They'd probably be okay, so long as the soil is not too compacted. You can also build up soil or mulch around the stem as they grow, which encourages more spuds without needing really deep soil. You can grow potatoes over winter (they like it cold and damp) when lots of other things won't grow - I don't know if that would reduce the chance of accidental double planting. In theory you can plant any potato (or cut one up - you just need a piece with an eye) but it's recommended to only use proper seed potatoes as they are generally disease free (and if you get a disease in your soil it is very difficult to get rid of).

We get rosellas in the front garden when the trees there are fruiting, and the odd honeyeater too, but they mostly seem to get on. Our old house was near a creek and the cockatoos and galahs going to roost there in the evenings could be pretty deafening.