In fact, virtually every modern piece of baseball analysis I've seen concedes that lefty pitchers are tougher on lefty hitters generally. That said, this is obviously only one factor. There are others, including the nature of the pitches, the type of swing, certainly other factors that make a pitcher tougher on particular hitters.
And I'll go even further with it. Sometimes a player seems to defy these concepts -- only eventually to fall into line. Ryan Ludwick is an example. He's known as a righty who hits righties. But recently he has been better against lefties, reverting to the typical form.
Even the great hitter Joey Votto has done better against righties of late as I recall, although in 2010 (without checking) he was just incredible against lefties.
In Parra's case, I just assume the Reds have viewed him and now that he has moved to the bullpen think he will be tough on lefties. As a reliever, rather than a starter.
I fully expect that they signed Parra intending to keep him, that they like the additional lefty in the pen, and will keep him. Which is another reason I expect Jose to depart.