Well, I've done the peat pot thing, or in my case the brown recycled paper pots from poundland ( the bigger ones ) , for years with consistent great results. Your own newspaper pots will do just as well of course.
Yes, you really have to be on the ball, and as soon as you see germination in your greenhouse or wherever, plant out the pots within a couple of days. If the weather is set fair at the time, as soon as I see one or two germinate, I plant them all out toute suite as the rest can't be far behind. Then water them as they really don't like to dry out. Just to be super duper extra careful, and probably isn't necessary if nice and wet and beginning to decompose already, I rip the bottom of the pot off before I plant. Oh, and I dib a big hole with a long spike first and fill with a handfull of compost before planting out each one. Sounds more trouble than it actually is and I can get out, say, 70 parsnips in a half hour. This way I get a 90% straight big parsnip return with next to no forking.
One more thing, with parsnips being notorious for their no shows some years when sown direct ( not liking to dry out at all in any way ), early pot sowings in the warm ( house or greenhouse ) give you plenty of time to resow should your first seed prove troublesome for any reason.
I wouldn't do the tipping out/individual transplanting thing with parsnips, but I do indeed do it with my early crop of carrots in trugs in the greenhouse, works a treat. Tease out, dib with a pencil, insert, and away you go.