Why is it that God the Father is alone capitalised in the third person? Usually Christians write He, His and Himself with a capital H when referring to God the Father. I'm not aware of this practice for either Jesus or The Holy Spirit.

Why the distinction?
Depends on the style guide, but it is a convention to capitalize names of God.

The Chicago Manual of Style does not require it, but does acknowledge it.

If a person doesn't capitalize He in regards to Jesus or the Holy Spirit, it is probably just an unintentional slip. 

Alternatively:


They could be Nestorian, and deny the divinity of Christ.
They might be Unitarian or Agnostic, and just not be sure.
Their Shift key might be possessed.
They might be anti-German, and thus hate capitalization on principle.


Or


They might have just screwed up.


"Never to attribute to malice where simple incompetence will suffice."