I've never tried this but I've read that one way is to use a pinch bar (crowbar) to make a hole a little deeper than you hope your carrots, parsnips etc will grow, wriggle the bar around until it makes a hole wider at the top than the bottom, then fill with compost or other fine soil. Sow a few seeds into each one, then thin to one - use the thinnings in the kitchen
Another way I have tried is to grow your carrots in a container of good light soil, maybe a 3gal bucket, with holes drilled in the bottom. You can pack the carrots quite closely; I have had enough for the whole summer from one of the big polyprop square containers you can get from, I think, Marshalls. Choose a variety which won't grow too long - I chose Nantes types mainly.
In a few years time, when you have been adding loads of manure and compost, sand and grit, your clay may start to improve. I grew up on Norfolk moraine clay, but my Dad improved the fields with copious amounts of manure, and his veg garden with all sorts. The soil in there was lovely, and he could grow anything. As well as cultivating the top, you need to pay attention to drainage as well.