Parkinson and Brown have joined as temporary replacements, while Lawrence is a supplementary pick
Matt Parkinson is set to play in his first overseas T20 competition after being signed by Peshawar Zalmi in the Pakistan Super League.
He has joined as a partial replacement for Saqib Mahmood while the fast bowler is on international duty in the Caribbean, where England are playing five T20Is.
Liam Livingstone will also miss the beginning of the tournament for the same reason, with Pat Brown – who has struggled with injuries since making his England debut in November 2019 – replacing him, also at Peshawar Zalmi.
Gloucestershire batter Ian Cockbain, who has impressed in the Big Bash after replacing Rashid Khan at Adelaide Strikers, will join Karachi Kings as a replacement for Tom Abell, who is out of the competition after injuring himself early on during a stint with Brisbane Heat.
Dan Lawrence, meanwhile, has signed for Quetta Gladiators as a supplementary pick. Lawrence, the Essex batter, was an unused squad member during England's Ashes series defeat.
How long he spends in Pakistan may well depend on whether he is included in England's Test squad for the upcoming series against West Indies in March.

Pat Brown is injury-free afte a difficult period caused by back problems (Nathan Stirk/Getty Images)
While this will be a first for Parkinson, who has featured four times for England in T20Is, Brown has twice been due to play franchise cricket abroad – for Melbourne Stars and Sylhet Sixers – but on both occasions was ruled out with back injuries before bowling a ball.
Speaking to The Cricketer earlier this month, Eoin Morgan challenged new death bowling options to emerge following England's defeat in the T20 World Cup semi-final against New Zealand. Brown seemed like he would be one of those when consecutive excellent T20 Blast campaigns saw him force his way into the international reckoning.
Since then, however, he has struggled to remain fit, playing through the pain barrier in 2020. He missed the Blast in 2021 as he worked his way back to full fitness, before returning in time for The Hundred, where he played for Birmingham Phoenix.