How do I delete most of my reminders on the iPhone?
I am not aware of any iOS or web app that will allow bulk deletion of iCloud reminders but that is something that presumably could be written.

The only mechanism I am aware of that will bulk delete reminders is the Automator app running on OS X.

Here is a screen shot of something that would find all reminders older than 20 days and delete them summarily.



You can see I searched for reminder and chose a task to select them and another to delete them. DO NOT RUN THIS unless you are prepared to lose all your reminders or test things on a second iCloud account.

For testing, I would add a second condition and watch the script run before letting it loose on a thousand reminders. I'd hate to be the one to find a bug in the time logic of the Find Reminders action by working on live data that's not backed up (and I don't know a good way to back up the reminder data and have it re-syced back to the cloud easily).