“A little history of British Gardening”, Jenny Uglow says of the medieval peasant:
To add to the grain for bread and the beans and peas … which came from their strip fields, villagers grew cabbage and kale, onions and garlic and leeks”.
There are references to turnips and parsnips being grown in the monasteries; but not to carrots.
I don’t think we should underestimate the technology available; they knew how to make fishing nets and grew hemp for clothing so it’s not impossible that the villagers netted their cabbages. Or perhaps (without the distractions of education or iPads) they sent the children out to pick off caterpillers and sit in the fields to scare off birds?
Perhaps the were growing things that were relatively pest free; no reference to carrots, perhaps carrot fly was a problem, and of course no concerns about blight as no potatoes or tomatoes until much later.