As I understand it, parsnips were the staple root vegetable in the British diet before the potato found its way to our shores. Common people in centuries past would not have had the means to roast them, so boiling would be how they were cooked much of the time. Ovens were found only in the homes of the better-off, or at the local bakery.

I should think the song is alluding to this in an attempt to suggest that it's older than it is.

Marje