After 74 from England's Moeen Ali, the 19-year-old England Lions batsman made 103 not out in an unbroken stand of 135 with wicketkeeper Ben Cox (59 not out).
Resuming on 336-5, the hosts reached 380 as Jack Shantry took four wickets, Hamish Marshall finishing on 135.
The visitors rallied after losing Daryl Mitchell first ball to post 297-5.
Worcestershire still trail by 83 runs with five wickets in hand, but they look a lot more comfortable than they did on 162-5 when, having lost captain Mitchell to the first ball of the innings, they were still 69 runs short of avoiding the follow-on.
World Twenty20 runner-up Moeen, in the first of three scheduled Championship games for his county, shared a stand of 120 for the second wicket with opener Brett D'Oliveira (40).
But, after hitting 14 fours and a six, he was caught at the wicket attempting to cut, beginning a mini-collapse of four wickets for 42 runs before Clarke, already tipped as a future England player, steadied the ship with Cox.
Earlier, Kieran Noema-Barnett had flicked Shantry's third delivery of the day to square leg to become the first of the morning's six victims, without adding to his overnight 84.
Gloucestershire captain Gareth Roderick:
"We had a very good first day, but all credit to Worcestershire. They played well.
"When we had them five down, we had a real chance to go on. But, once again, credit to them for the way they batted and now we have to front up.
"We need a good first hour and then take it from there. Hopefully the first session will be more fruitful than it was today."
Worcestershire director of cricket Steve Rhodes told BBC Hereford & Worcester:
"It was tough losing Daryl so early when we batted, but Moeen and Brett batted really well and then the young lads Joe Clarke and Ben Cox came together and were excellent.
"It's great to see two young England lads playing like that. There are a lot of overseas players playing here, but to see young English lads to the fore is very pleasing.
"Joe is a fine prospect. He likes to score and put pressure on the bowling side. He bats with a nice tempo and has a habit of putting away the bad ball."
Highly-rated Worcestershire teenager Joe Clarke made his third first-class century to keep his side in contention against Gloucestershire at Bristol.