Roots, Mulch and No Dig

I practice a sort of No Dig, forking out mole or vole runs as needed, which can sometimes be a whole bed, but in general just mulching with compost and grass clippings to keep down the weeds and put goodness back in.
However, I've always left that year's root patch unmulched not wanting to add too much goodness, but this leaves the soil exposed to the heat and rain as well as passing weed seeds. Now I'm thinking I should be mulching here, too. 
I only grow parsnips and carrots, neither particularly successfully, in the roots rotation.
Any thoughts?

Charles Dowding of no-dig fame adds mulch to his beds where he sows carrots and parsnips, just like every other bed, according to his videos. I'm growing some carrots in a recently compost-mulched bed at the moment, so my firsthand experience will be a while yet in coming CD seems to think that the compost is ok for the roots as long as it's not forked into the soil but applied as a mulch.

Could you cover the roots bed with cardboard or summat to protect it from sun/rain/weed seeds?

I don't dig unless it necessary to dig out perennial weeds or to plant something - never have done as it always seems like a waste of energy.
I don't have enough compost to follow CD's practices and I won't be buying it either. Buying compost to cover the ground makes gardening too expensive for me - said the Moneyless Chicken.

I've just hoed and sowed some snips so will spread some clippings once I have some seedlings to mulch around. I'll certainly not be using bought compost to mulch either!!
Thanks both.