As the owner of a sub-site, you have full control. You can create a list and get to list settings. But when you click Save List as Template, you get an error.
The reason is that list templates are stored in the List templates gallery—which is a document library—on the root of the site collection. If you don’t have Full Control there, you can’t save to the gallery. Thus, you can’t save a list as a template.
You have two options. First, you can obtain the necessary permissions to the root of the site collection. If that’s not possible, then your site collection administrator will have to create the list template for you. Perhaps with enough requests, you’ll get the permissions you desire.