On my main site
example.sharepoint.com I can add webparts and save it like a template but on a subsite
example.sharepoint.com/sites/project1 I dont have neither of those options. In admin center I have allowed scripting.
If I am not mistaken, what you are referring to is the Galleries section of the site settings where "Web parts" and "List templates" are options. These options are only available at the root of the site collection because that is were list templates and web parts are stored.
You can still save list templates from a subsite, but to upload them for use on a site collection (including the subsite), you must navigate to the root of the site collection, go to Web parts or List templates, and upload them there. Than if you navigate back to the subsite, you will see they can now be used.