By default ticket creators can close their own tickets, but this can be disabled so that people can't close their own tickets. This is so you have complete control over the ticket, or stopping people from spam creating of them.
To disable this, go to the panel (https://panel.parent.gg/)
Parent.gg - Login
Select your server and click "Command Settings"
Once you have clicked "Command Settings" click "Close Command" and open up the option and find the below.
All done, ticket creators can't close their command. If you're going to test this, if you have a staff role this will overrule this setting, so we would recommend you test this on someone who hasn't got staff permissions.