is the account IMAP or POP? That must be resolved as the answer depend on which is configured.

Right click the account in the folder pane and select settings. Click on server settings for the account once the account settings open. At the top on the right will be shown the Server Type:

It's IMAP.

I want to be able to delete mail from Earthlink while keeping it on thunderbird.

That won't work with an IMAP account. You'd need to create a POP account. See https://support.mozilla.org/kb/faq-changing-imap-pop