You're using a primary and child cox.net account and see both are receiving duplicate emails? Are you using a client with a pop server or imap? If you're using a client for email, which one are you using?
If you're using pop, then you may want to switch to imap. https://www.cox.com/residential/support/cox-email-server-settings.html is a link for server settings if you're looking to change from pop to imap.