Does Bitcoin limit the number of parallel connections from the same IP? Is there a measure in place to prevent an adversary from connecting to a Bitcoin node many times in parallel, thus saturating its open connection slots?
The measure is that the node running bitcoin removes the cap to the number connection slots.  The cap for connection slots I have found is set to no limit when running a bitcoin service.

Sorry I stand corrected, back in 2011 looks like a cap of 125 connections was introduced Is there a connections limit on bitcoind? 

in the net.cpp file  https://bitcointalk.org/index.php?topic=52772.msg629593#msg629593

but this has since been changed and I dont think there is a cap anymore... Looking at the file looks like this is the area you are interested in: https://github.com/bitcoin/bitcoin/blob/master/src/net.cpp#L1081
and the values set here https://github.com/bitcoin/bitcoin/blob/master/src/net.h