Parsnip seeds will take a fair amount of frost without damage. Not saying they will germinate though, they are notoriously prone to being sold already dead. At least if they have not come up in a few weeks time you can sow again.
We often do not sow until later in the year, April even as we are rather cold here.

Parsnips often give variable germination results and although seed packets and gardening books will recommend sowing at the end of February, if the ground is wet and cold the seed will tend to rot off. I would wait at least until the end of March at the earliest. Even under good growing conditions, the parsnip seed takes quite a long time to germinate. It is also wise to buy fresh seed every time. I would also suggest that you sow the seed in groups of about 4 seeds along the row. When the ones that germinate get about 1 inch high, pull out the weakest one. Then as the remaining ones become bigger again pull the weakest. This way you end up with a continuous row of parsnips with no misses.

When I first started we always sowed Parsnip seed first, then Shallots.

One year I tried sowing Parsnip seed on wet paper on a tray in the warm airing cupboard. It germinated well. I kept them damp and cut out the little seedlings and planted them in the garden in March. Will do that again this year.

Been there, done that. Also got them to germinate in wall paper paste and sowed them, still in it, into drills in the ground.

Cheers all, I have sowed quite thickly so as to thin out the weaker seedlings, will cover them in fleece and see what appears in March, hopefully the damage isn't already done. Sounds interesting doing in wallpaper paste and wet paper, is it not very time consuming transplanting them after? The bed I am using this year is about 3 metres long so could imagine that would take quite a while... Does the foliage appear first on the top of the paper / paste or the root underneath?

With the paste, all I did was put the germinated seedlings (one root and a tiny leaf) still in the paste, into plastic bag with the corner snipped and piped (like thin icing) into a drill and covered the paste over with a thin layer of soil.
Either way, you need to have the soil warm and the seedlings need to be only just germinated so you do not damage the root in any way.

The airing cupboard was not a good idea. The paper dried out quickly and when I added more water the seeds floated off. Should have a good crop of Parsnips in the airing
cupboard ! Have now removed the tray to the draining board on the sink, leveled with a teaspoon at one end.
Can keep an eye on them there. Will also try the wallpaper
paste method, be a good way to use up the left over
unused grains.