OED has 1597 for the earliest citation of pissabed for dandelion. Dandelion itself is dated back to 1513, not that much earlier.

And, le Trésor de la langue française informatisé gives late 15th century (ca.1483 by another source) for pissenlit in French. That's a gap of almost five and a half centuries between the Norman conquest and 1597. There were a lot of French loanwords coming into English in the 15th and 16th centuries. And pissabed is not a loanword, but may be a loan translation (calque). Because other Romance languages have a similar word (e.g., Spanish meacama) and given its late citation date, I think it's more likely the English probably borrowed and translated it from early modern French. OTOH, dandelion was probably borrowed from Norman or Anglo-Norman.

Ceci n'est pas un seing.