Justin Martyr said in the second century that Mithras's caves was a demoniacal imitation of the tradition that Jesus was born in a cave. Scholar Ernst Benz "has shown conclusively that this Christian tradition does not come from a dependency on Mithraism, but rather from an ages old tradition in Palestine itself of holy shrines in caves." There is no doubt that the Christian tradition does not stem from the Mithraic account. Mithras being born on December 25 is not a parallel. We don't know the date Jesus was born. The earliest date celebrated by Christians was January 6-in fact, it's still celebrated by many churches in the East. Of course, December 25 is very close to the winter solstice. This was the date chosen by emperor Aurelian for the dedication of his temple to Sol Invictus, the god called the 'Unconquerable Sun.' Mithras was closely associated with Sol Invictus; sometimes they're depicted shaking hands. This is apparently how Mithras became associated with December 25." The day that date became Christmas for Christians seems to be in 336, a year before the death of Constantine, the first Roman emperor to embrace Christianity. We know that before his conversion, he worshiped Sol Invictus.
We know for sure that Constantine made Sunday, or the Lord's Day, an official holiday, even though Christians had already been observing it as the day on which Jesus was resurrected. So it's conceivable Constantine also may have appropriated December 25 for the birthday of Christ. We know that Christian emperors and popes suggested that instead of simply banishing pagan ceremonies that they appropriate them for Christianity.