The Parachure Engineers in 1 & 6 AB Divisions were termed Squadrons for some reason even though they were Companies in the rest of the Royal Engineers. Your dads BD jacket appears to be the post war version with the open neck and tie so was probably taken post war. He looks a bit like Cliff Richard.
1st Parachute Squadron RE
We had a lad join 10 Para in 1992. He had originally joined the Royal Anglians as a boy soldier at IJLB Oswestry then served with one of their battalions until he transfered to the RE and joined 9 Para Sqn RE serving with them about 1978 to 85 including the Falklands war in 1982. He then went to JLR RE as an adult instructor as a Corporal until he flounced in about 1987/88.
When he joined us he had just finished working in Kuwait clearing mines. He had paid for a house in London with the money he had earned and complained that he was down to his last £70,000. He wasn't EOD and got the job purely on his B1 Combat Engineer experience.
Apparently Royal Ordance had got the contract from Kuwait to clear all the minefields and then contracted it out to the Royal Engineers in the best tradition of Tory government back scratching their mates. RO was in the process of being privatised and sold off at the time - hence the SA80 saga of Value-for money, or doing it on the cheap as it was more commonly known to ensure RO had full order books.
I think it was 50 Field Squadron at Rippon was due to go but the chaps got a bit miffed about risking life and limb to bump up shareholders dividends and Directors profit and it hit the front pages of the papers. The government bottled it and RO had to shell out the appopriate rate for contracters to do the job.