The issue here is that parallel is not properly formatting the rsync command. The rsync command expects a source and a destination, but the way parallel is passing {} is incorrect.
Ensure {} is prefixed with the remote hostname properly. Modify your command as follows:
ssh -o ConnectTimeout=10 -i /home/ec2-user/.ssh/remote.pem [email protected]
-q -t 'find /var/log/ -type f -print0' |
parallel -j 5 -0 rsync -aPvv -e "ssh -i /home/ec2-user/.ssh/remote.pem"
[email protected]:'{}' /home/ec2-user/local_log/