This is really a job for Andy but I believe he's busy doing intensive on-the-spot research for us around the bars of Symi at the moment, meaning that a reply from him could take a little time so I'll give it a go for you.
As far as I know the Blue Star Diagoras (or Paros) only calls at Symi on Wednesday and Friday. Their online booking system is offering Rhodes-Symi crossings on June 12th, 17th and 19th. Nothing on the 15th. Perhaps you are looking at the wrong boat? The Blue Star 2 leaves Rhodes at 16.00 on Mondays but goes directly to Kos and then on to Santorini and Piraeus, not calling at Symi.
Sadly, Monday isn't a good day for ferry connections to Symi with only morning sailings