To take part in an Amphibious Landing, I would prefer the Marines. To take part in an Airborne Assault, I would prefer the Parachute Regiment. When on the ground I would probably prefer an Armoured Division.
Because there seems to be a little bit of a confusion here, I'll explain.
The British Army is different from the Royal Navy and the Royal Air Force in the fact that the RN and RAF were created under Government/Royal consent (hence the 'Royal' part), these two organisations were created as a whole.
The British Army on the other part is a collection of local Armies (originating from times when regions would have their own Army rather than a national Army), for this reason it is not the 'Royal Army'. However, you will find many of the Regiments and Arms will have names like 'Royal Engineers' and 'Royal Anglian Regiment' because THEY were created with Royal Consent.
Therefore... the Royal Marines are a part of the Royal Navy (the Navy's infantry), while the Parachute Regiment is a Regiment/Organisation in itself but obviously answers to the higher authority of the British Army and is a part of the British Army.
As for who is better equipped, I say it depends on the situation. Yes, they are both demanding, that is because they are both spear-head battlegroups. Marines much take the land or else; Paratroopers much touch down safely, regroup and establish a path for non-Airborne task forces.
Inaspin said it exactly, I just like the sound of my own voice more.