Very nice jacket!
The jacket is from a member of 1st Battalion, as Brigade Piron correctly points out. This is denoted by the maroon "Pegasus" arm and metal epaulette badges. 2nd and 3rd Battalions have their own unique metal badges for the shoulders, only 1st Battalion has the maroon arm badges.
The core structure of the Para Commando Brigade was as follows:
1 Para Battalion
2 Commando Battalion
3 Para Battalion
Various support units
As of 2003's reorganization the Brigade is gone and instead the three Battalions exist on their own. Support units were either disbanded or added into the Battalions.
The Commando badge on the right arm is a qualification badge, not a unit badge. This is called the Commando A award. Commando B is rarely seen on a uniform (it's the lesser of the two). Any member can have this certificate, not just those assigned to 2nd Battalion. The jump wings are Parachute A. As with Commando A, it is the superior grade. Parachute B is not generally seen on uniforms.