Sure - it's different in terms of frequency of 0 days rest usage. And as I noted, some of those BOS appearances had higher pitch counts.
But is it significantly different overall? Between April 4 and May 4, the Sox called on Brasier 13 times. Between June 21 and July 21, LA called on Brasier 12 times. And yes, that includes the ASB.
Or go with the whole season thus far:
March 30-May 14, BOS. 20 games, 21 IP, 372 pitches. prorates on a 162 game season to 68 games, 71 innings, 1265 pitches.*
June 21-Aug 28, LA. 27 games, 28 IP, 361 pitches - prorates on a 162 game season to 68 games, 71 innings, 909 pitches.
So I don't think the argument can really be that he was used more by the Sox, or less by the Dodgers, and something starkly different resulted in Ryan Brasier, the same pitcher, pitching better simply because he was well-rested.
The difference is that LAD coached him into being a better pitcher.** And then used him at about the same rate as the Sox did.
* Although if we take out Cora's final throw-him-to-the-wolves game, he'd have 19 games, 18.2 IP, 330 pitches, prorating to: 68 G, 67 IP, 1181 pitches. Here, I think it's fair to consider the pitch count numbers without the last game. They clearly were musing DFAing him to use him like that.
** Which as an aside results in a somewhat lower pitch count. So I'm sure he feels marginally less drained.