The FIDF is part of the garrison but it's a civilian defence force recruited from the local population, there's also a regular British Army infantry company based on the islands 24/7/365.
Note: A very handy figure people toss around is "Oh there's like 1000 british infantry" on the island, which isn't correct. There are over 1000 military personnel but in terms of frontline infantry there is still one company, the rest is RAF personel (Operating + maintaining 4 Typhoons, one VC-10 tanker and a C-130 i think
+ a couple of SAR Sea Kings) and general logistics people, Rapier AA battery operators, HMS Clyde + HMS Edinburgh (T42 destoyer) is relieving HMS Dauntless in the region.
In February of this year the company was provided by 2nd Scots, it's probably changed by now though.
There's not much point in deploying something like the Paras or something highly specialised, the tasks the Army would be expected to do would be well within the capabilities of a regular British Army infantry company.
Most of the key land units are still stationed in Afghanistan, a regular light infantry company is perfectly acceptable now. Considering what there is ~300 miles away it'd be fine. Logistically it's easier to deploy a light infantry company abroad than a mechanised company and I expect most of the air assault units - rightly so - are tagged for 16 AAB as the current rapid reaction brigade.
BBC puts the numbers at a regular company ~100 men with ~200 reservists that make up the FIDF. Interesting to note that any images of the FIDF show them armed with the Steyr AUG, if it's rugged enough for the Aussies then it's rugged enough for the FIDF