I doubt statistics are available on the quantities of different crops cultivated in mediaeval times and earlier, so we will have to do with lower quality evidence.

Wikipedia says
"Until the potato arrived from the New World, its place in dishes was occupied by the parsnip and other root vegetables such as the turnip." Unfortunately it does not source this comment. In fact practically nothing in the Wiki article on the parsnip is sourced. But note, it does not say that parsnip was a staple, but rather that root vegetables in general took the place that potatoes have today.

I would dispute your assertion that parsnips have all the advantages. Rather potatoes have all the advantages (except possibly taste, though I would say a dislike of parsnips is a lot more common than a dislike of potatoes). My understanding is that the yield of potatoes is larger than the yield of other route vegetables. Also being more starchy than other root vegetables you can grow in cool climates, they are more nutritious - typical calorific values per 100g from Wikipedia are 55 for parsnip but potato is 77Cal, which is quite a lot more.

So I would suggest potatoes became the major root vegetable in temperate climates because both the yield and nutritional value is typically substantially higher. That is why Irish people living on small pieces of land grew little but potatoes, and hence starved when the blight hit, because they were mainly growing just one variety, the Lumper.