As many here will tell you, the short answer to your question is to practice light attack weaving: hitting LA and canceling the animation for it by using a skill immediately after. All the sites you're getting your builds from will also have guides to LA weaving. It takes a while to build up muscle memory for it but once you do, you'll be able to hit far over 15k parses.
There are a few more components to account for though. When people and guides post parse, they are using the Iron Atro trial dummy, which provides buffs and debuffs that will automatically increase your dps. Starting on a 3 mil dummy is fine to practice, but once your weaving is consistent, you'll want to look into practicing on a trial dummy. The easiest way to access one is through a guildmate/guildhall. There is also the new "Harrowing Reaper" dummy for sale now if you're willing to drop 6k crowns for it.
A few other things: I personally wouldn't recommend Tinkerer's as a stand in for Medusa's, as the proc for that set drives tanks crazy and proc sets in general will be nerfed next week for mag builds. Julianos is a quick stand in that requires zero farming, but there are plenty of other sets too. Note that golding out weapons is much more important than it is on armor, while golding jewelry has even less of an effect. Divines is helpful, but not as important as nailing your rotation and weaving, and I'd probably prioritize transmuting weapons, then jewelry, then armor.
Final point: keeping up dots and buffs is another important part of dps, and if you're on pc, I'd recommend using an addon that helps you do so. I personally use Action Duration Reminder, but there are many out there. And Combat Metrics is an extremely useful addon for reviewing your rotation after a fight.