You should have sown them directly into the soil!

Most (not all) root vegetables do not take kindly to transplanting as you have found!

I agree with Tg sow your carrot and parsnip seed where you want it to grow in a fine, sandy soil. Thin out as needed and protect from carrot root fly.

Most of the plants which are ROOT crops have a long tap root which must not be disturbed even in very early growth. The only type of carrot you might try is the 'round' type which grows very short.
If you really want to start the seed in the greenhouse, which is really not necessary, you can germinate it in damp vermiculite and then very carefully move the tiny seedling to its permanent position in the garden and drop it into a little hole also with vermiculite in it.
I now find it is easier if you have difficult soil where carrots and parsnips don't seem to germinate, to dib small holes in the well tilled ground with the end of my finger, fill this carefully with perlite or vermiculite, drop a seed or two in the hole and cover with another sprinkling of the same. This keeps the seed nice and moist especially if you cover with a polythene sheet until germination has started. Parsnip seeds can take many weeks to germinate, and carrot seed ten days or two weeks depending on temperatures. It is vital to keep seeds like this from drying out but not so wet that they rot.

Bertie says it all

I didn't mention 'chitting' a system I often use with Parsnips as they are notoriously bad germinators.

The reason I didn't was I didn't know how experienced a gardener you are and where you live.

Timing is crucial so you have got to do it a time when you know when you can plant them out into their final quarters.

If they are allowed to remain in the cells too long the tap root will be hindered then you end up with weird shaped Parsnips.

If you decide to use this method then my advice would be to sow a couple of weeks before the time you know you can plant out, in this way the roots will be only a few millimetres long and won't be touching the bottom of the cell when you finally transplant them out (I hope that makes sense!)

This link has a few slides on it that show you how I chit mine;
http://www.thegardenersalmanac.co.uk/Data/Parsnip/Parsnip.htm

With carrots I find it is better to sow a latish variety rather than an early variety as the ground should have warmed up sufficiently to germinate them.

Early varieties often get hit with a cold snap and they abort!

If I want early carrots I use the thinnings of my late variety and get the best of both worlds!

http://www.thegardenersalmanac.co.uk/Data/Carrot/Carrot.htm

I'm afraid that I belong to the old school and too will always sow my carrot and parsnip seeds into the ground. I also belive that you can break the tap roots and the check in growth from transplanting the seedlings gives little advantage. Even sowing into the ground a little later the plants soon catch up with the transplanted plants. Parsnips are best sown later in stations (3 or 4 seeds together) rather than in continuous rows and the weakest seedling pulled out, eventually leaving the strongest seedling and making a full row.

thanks for all your replies. I think next year I will sow direct in to the large tubs that they will grow in, What is the best feed for root veg?