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."

The flight search engine website reported revenues of Â£120m for 2015, up 28% on the previous year.
The company saw gross bookings of $11.2bn, up 49% on 2014, with visitors to the website up 48%. On average, 50m people used the website each month.
Company founder and chief executive Gareth Williams said the company had seen "excellent growth".
He said: "2015 was another strong year for Skyscanner, showing solid progress in our strategy of expansion beyond our home market, into mobile and beyond flights.
"Our first-half focus on investing in our product drove momentum for the business in the second half. We also saw excellent growth in travellers planning their trips on Skyscanner all around the world, especially on mobile and in the important regions of Asia Pacific and the Americas."
The Edinburgh-based company saw a rise in the number of people using its mobile app, with 40m downloads and an increase of 24% in mobile bookings.
Particular growth was recorded in the Asia-Pacific region, while visits in China were up 67% after Skyscanner bought a Chinese search firm in 2014.
The company opened new offices in London and Sofia as well as expanding existing operations in Budapest and Miami, increasing its global staff to 770.

Gloucester have played away from their Meadow Park home since a flood in 2007.
In September, planning permission was approved to permit initial work on building a new ground for Gloucester.
The University of Gloucestershire-based All Golds, who are in the third tier of rugby league, currently play in Cheltenham but want a permanent home.
All Golds president Lionel Hurst told BBC Radio Gloucestershire: "We have met with those who own the club.
"We can see that it makes considerable sense. We will need to have more meetings.
"We are a long way from getting over the line on this, but watch this space. We are constantly seeking our own permanent base for the club.
"Rugby league and football are very good friends throughout the land. It would be a very significant partnership if it happened."
Worcestershire batted themselves to a good position on day two against Glamorgan in Cardiff.
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Scottish-based travel firm Skyscanner has reported double-digit growth for the seventh consecutive year.
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National League North side Gloucester City are in talks with rugby league team Gloucestershire All Golds about a potential groundshare.