Father's lab coat is a unique lab coat with a white collared shirt under a green knitted jumper. It also provides a bonus of +2 Intelligence when worn. The coat can also be equipped with ballistic weave but cannot be worn with pieces of armor over it.
It can be obtained after the completion of Nuclear Family. After the final cutscene and the ending screen, speak to Allie Filmore at any point afterwards to inform her of Shaun's death. She will then proceed to tell you that the director's quarters are now yours. After some time, Father's lab coat and the orange Institute division head coat will spawn on a desk on the upper floor of the director's quarters.
Alternatively, if you have no intent to side with the Institute, simply kill Father and loot the coat from his body. This will cause the player to be banished from the Institute, so be sure to finish whatever you need to do with the Institute before killing Father.