His career record against them was 28-13. From 1955-61, it was 27-10 with a 3.06 ERA while his ERA against everyone else was 3.42. But did he really pitch better or differently against the Yankees?
Let's start with strikeout-to-walk ratio. In those years, Lary's was 1.62 against non-Yankee teams (I included HBP and took out IBBs-all data from Retrosheet). Against NY, it was 1.71. That may seem consistent with the "Yankee Killer" nick name, but over those years the Yankees themselves had a 1.43 ratio while the rest of the league had 1.32. So the typical pitcher had a strikeout-to-walk ratio that was .11 higher against the Yanks than everyone else. Lary was .09 better. So he was doing just about what other pitchers did.
Now HRs or HR rate (I use HRs divided by PAs with IBBs taken out). Lary allowed the Yanks a 2.6988% while he allowed the rest of the league 1.75%. So the Yanks did about 0.948 percentage points better against Lary than the average team from the rest of the AL. But that is just about normal. Over these years, the Yankees had a rate of 3.0097% while the rest of the league had a rate of 2.162%. The Yankees were about 0.848 percentage points better than the league average. So again, Lary's relative performance vs. NY is about what it was for other pitchers.
What about other hits? Lary's non-HR hit% against NY was .199 while against other teams it was .217. So that is a fairly big improvement. Some how he was better at preventing hits against the Yankees than he was against other teams. The Yankees themselves had a .205 rate while the rest of the league had .204. So the typical pitcher allowed more hits (but not alot more) to the Yankees than they normally did.
So it seems like the one thing that Lary was good at when he faced the Yankees was in preventing them from getting singles, doubles and triples. But the difference was only .018. Over, say, 36 PAs per game, that is just .648 hits. The run value of those hits is about .55 (the weighted average of the linear weights values that Pete Palmer established). So that makes a run value of .36 (interesting that that is just about the difference between his ERA against other teams and the one he had against the Yankees, 3.42 vs. 3.06).
The Tigers did score 4.93 runs per game in his starts against the Yankees from 1955-61. They averaged 4.61 runs per game overall. So the hitters rose to the occassion to support him. And maybe the fielders played a role in lowering the rate of non-HR hits he allowed. So it is possible that Lary became the "Yankee Killer" due to the aid of his teammates.