Your first sentence, 'Potatoes are an important staple crop in many parts of the world.', is fine, and needs no correction.
The third sentence, 'The potato is an important staple crop in many parts of the world.', is also fine.
The second sentence, 'Potatoes are important staple crops in many parts of the world.', isn't quite right. The plural 'crops' seems wrong. Rice is a crop, wheat is a crop, and potatoes are also 'a crop' - singular. The fact that 'potatoes' is in a grammatically plural form does not alter the fact that it's a single crop. If you are talking about different varieties of potato, then - at a pinch - they could be referred to as separate 'crops', but that doesn't seem to be the case here.
I hope that's helpful.