The accepted answer does not resolve the question of which is more correct in the grammatical sense although it does provide excellent information about which is more correct in a legal sense.
To specify the grammatical sense of which is more correct, one could rephrase the question:
If someone asked me for permission to use my work, should the usage statement I provide them include "used with permission" or "used by permission"?
The meaning of "by" in this case refers to the means by which something is achieved. "I reached the top by climbing the dead bodies of my enemies," for example. In our case, what has been achieved is the mere inclusion of our work within the work of another. As such, is the means for achieving permission to use the work the permissions themselves as, "used by permission," would suggest? No, permission is a thing which is acquired through the effort of asking for permission.
So a statement describing what is actually the case in our situation would be, "used with the permissions granted to me by asking the owner for permission."
Therefore, if by were the correct word here, rewriting our statement for brevity would yeild, "used by asking for permission," which is clumsy and no fun.
Or, to use with, used with permission, which is both succinct and correct.