I'm fairly certain the comment you linked to is thinking of a declaration Edward III made in 1363:

Whereas the people of our realm, rich and poor alike, were accustomed formerly in their games to practise archery – whence by God’s help, it is well known that high honour and profit came to our realm, and no small advantage to ourselves in our warlike enterprises…that every man in the same country, if he be able-bodied, shall, upon holidays, make use, in his games, of bows and arrows…and so learn and practise archery.

(I couldn't find the original declaration in full text, but you can find the excerpt in Tools of War: History of Weapons in Medieval Times by Syed Ramsey or English Longbowman: 1330 – 1515 by Clive Bartlett).

As you can see, it says nothing specifically about longbows though I can't find any sources talking about the types of bows peasants used for hunting or sport. One thing to note, in the declaration of 1363 and again in 1388, Edward III banned football (this askhistorians post goes into great detail on the various football bans through the centuries) in favor of archery as a sport for peasants.

So peasants were required to practice archery though they were not specifically required to practice the longbow.