Their 196-run first-innings lead came mainly thanks to Tom Kohler-Cadmore (102) and Ed Barnard (59).
Further contributions down the order allowed the visitors to reach 403 all out for five batting bonus points.
Glamorgan started well with Jacques Rudolph and Nick Selman putting on 74 for the first wicket, but they ended 141-3 at the close, 55 runs behind.
Worcestershire claimed three wickets for 34 runs to reduce Glamorgan to 108-3, but Colin Ingram (41 not out) and Aneurin Donald saw out the rest of the day.
Glamorgan bowler Michael Hogan told BBC Wales Sport: "It seemed to go through different stages at different times. It was a little trickier at times and then got easier as ball got softer. The boys toiled hard but it could have been a bit better.
"It was good to see Lukas Carey doing well. He has done a lot of work in pre-season and he has worked really hard on his fitness and his technique, so it is good to see him being given a chance and taking it with both hands.
"I'm fitter after a winter off, probably lacked a bit of bowling conditioning. I've had a few weeks off with injury so probably I got bit tired towards the end.
"We thought with the sun out after about 35 or 40 overs it got a bit easier, but they have got a good attack so we will have to absorb a bit of pressure. I think the boys can do the job and if the boys can build a good total we can hopefully bowl them out."
Worcestershire batted themselves to a good position on day two against Glamorgan in Cardiff.