Father's Lab Coat

Father's lab coat is a piece of clothing in Fallout 4 worn by Father.

Characteristics
Father's lab coat is a unique lab coat with a white collared shirt under a green sweater vest. It also provides a bonus of +2 Intelligence when worn. The coat can also be equipped with Ballistic weave to add Damage Resistance at an armor workbench.

Locations
Father's lab coat can be obtained after the completion of Nuclear Family. After the final cutscene and the ending screen, speak to Allie Filmore at any point after 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.

It's also available to be pickpocketed briefly after Mass Fusion and before completing Powering Up, perhaps as a result of a bug.

Bugs
A second copy of the coat may be acquired by killing Father and looting him, which may cause another coat to spawn on his body.