Well quietdandelion this has got a long way and nobody has answered, so I'll have a try for you, even though I'm English and that comic is mostly American in humour.
I'm not conscious of a phrase 'the pirate stage of a relationship' so I think the joke must be nothing more than that they've reached the stage of their relationship (marriage or partnership) where, to make life more interesting and exciting, they dress up as pirates. In the picture both are wearing pirate costume, down to the parrot on the shoulder of Mr Hemeshot.
It doesn't strike me as very funny and would have much more point if there were an expression about the pirate stage of a relationship. We'll have to wait for a American to give us that detail.