Reopen closed ticket with API

Thank you for posting your question in the Community! The Closed status is permanent and even via API can't be updated/reopened. You could create a follow-up request for each that references the closed ticket. However, that is something that would have to be done manually. 
 
You could use the API to create these follow-up tickets. The create_many endpoint allows for up to 100 tickets created per request. 
 
I would recommend creating a script that creates this new follow-up ticket with a canned response (maybe as an internal note if you didn't want it to notify the customer). You can include a dynamic statement that links to the specific closed ticket. You would need to have a list of all of the ticket id's for the accidentally closed tickets. Then you would get the closed ticket's specific data (requester, assignee, additional ticket field values) by looping through all of the ticket id's.
 
I know it's a bit of a long workaround but it would probably be the safest option to make sure nothing is missed.