The 23-year-old Lancashire bowler, who made his international debut against South Africa in February, was injured during field training on Monday.
The three ODIs, which will take place behind closed doors in a bio-secure bubble at the Ageas Bowl in Southampton, will kick off with the first game on July 30.
“An ankle injury has ruled out Matt Parkinson from the Royal London Series with Ireland,” the England and Wales Cricket Council (ECB) said in a tweet.
England has yet to name a replacement for Parkinson’s.
Parkinson has chosen no wicket in his two ODI appearances but has won five wickets in the two T20 internationals he played against New Zealand in November last year.
Ahead of the ODI series against Ireland, the ECB also announced that spinner Moeen Ali would be deputy to skipper Eoin Morgan.
“Moeen Ali has been confirmed as Eoin Morgan’s vice-captain for England’s three Royal London Series matches against Ireland,” the ECB said in a statement.
England will decide their squad for the series after the end of the two intra-team matches and a warm-up game between England Lions and Ireland on Sunday.