By the two birds Christ is denoted, both dead as man, and living as God. He is likened to a bird, because He is understood and declared to be from above, and from heaven. And the living bird, having been dipped in the blood of the dead one, was afterwards let go. For the living and divine Word was in the crucified and dead temple [of the body], as being a partaker of the passion, and yet impassible to God.
By that which took place in the running water, in which the wood and the hyssop and the scarlet were dipped, is set forth the bloody passion of Christ on the cross for the salvation of those who are sprinkled with the Spirit, and the water, and the blood. Wherefore the material for purification was not provided chiefly with reference to leprosy, but with regard to the forgiveness of sins, that both leprosy might be understood to be an emblem of sin, and the things which were sacrificed an emblem of Him who was to be sacrificed for sins.
For this reason, consequently, he ordered that the scarlet should be dipped at the same time in the water, thus predicting that the flesh should no longer possess its natural [evil] properties. For this reason, also, were there the two birds, the one being sacrificed in the water, and the other dipped both in the blood and in the water and then sent away, just as is narrated also respecting the goats.
The goat that was sent away presented a type of Him who takes away the sins of men. But the two contained a representation of the one economy of God incarnate. For He was wounded for our transgressions, and He bore the sins of many, and He was delivered for our iniquities.— From manuscript of writings of Justin.
Source: Miscellaneous Fragments from Lost Writings (New Advent)