I haven't seen this comic before so I'm not clear on the humour but I guess either to let his friend have his '15 minutes of fame' or else to tease him a bit for showing off (in the image the main character says 'Dakota' instead of 'America').

In this context 'to parade (somebody or something)' is to walk around with them in very visible or public way. To show off that person publicly. Hope this helps!