Resuming on 294-8, Worcestershire salvaged another batting point before being bowled out for 307.
Teenage red-ball debutant Jack Home smashed successive boundaries off Chris Rushworth to reach 29 in his maiden first-class innings.
But Rushworth then quickly got his own back, trapping him lbw before clean bowling the Pears’ last man, on-loan Amar Virdi, two balls later.
The Bears quickly lost Rob Yates, who was caught by Virdi at mid-on from a leading edge off Tom Taylor, but Davies and Rhodes saw their side through to lunch on 76-1.
The afternoon session began badly for the visitors when Rhodes was out to the first ball after a slightly delayed resumption, caught behind off former Warwickshire team-mate Ethan Brookes.
Brookes then added another scalp when he bowled Davies, before the Bears lost a third wicket in 10 overs when Sam Hain almost overbalanced trying work a ball outside off stump onto the leg side and saw Netherlands international Logan van Beek clip the bails.
Soon after that, the umpires intervened, for bad light. Rain followed – and the premature end to the proceedings was by then inevitable.