In canon, the assignment of eligible hearers to prophecies is done by Minesty workers. Specifically, the judgment that "the one with the power to vanquish the Dark Lord" refers to Harry, and thus that Harry should have access to the prophecy, was made some time after the recording of the prophecy, by a human. On the assumption that things work the same in the rational-verse, the fact that Lily and James could hear the prophecy isn't evidence of anything other than the interpretation of the Minestry worker who handled the case.