With rain interrupting play, Greg Smith and Michael Thornely shared a partnership of 93 to lead the hosts' bid to save the game. Thornely was out for 43 but Smith, dropped twice, was still there on 62 at the close with Leicestershire at 133 for one.
They are still 167 runs behind, with Glamorgan having declared on 442 for nine on Wednesday and then restricting their opponents' first innings to 142, but as more rain is forecast for the final day, Leicestershire will harbour hopes of holding out for a draw.
Rain also affected Worcestershire's home contest against Gloucestershire, which was on day two.
The hosts, who had bowled out Gloucestershire for 234 and got to 65 without loss by the close of play on Wednesday, moved on to 302 for three on Thursday, with Moeen Ali top-scoring by accruing an unbeaten 122.
Another match on day two and hit by rain was Essex's encounter with Kent at Chelmsford, where the away side, having resumed on 20 for one in response to 212 all out, recovered from a shaky start to close on 214 for five, with Darren Stevens on 91 not out.
On day one of their clash with Hampshire at Southampton, Lancashire won the toss, chose to bat and were on 252 for nine at stumps, Andrea Agathangelou having recorded an unbeaten knock of 93.
David Balcombe took five wickets for 92.