The Sky Blues, who sacked Aidy Boothroyd on Monday, went behind when Rodriguez lobbed Keiren Westwood.
The Republic of Ireland Under-21 international played under new U's boss Aidy Boothroyd at Vicarage Road.
King was dismissed for a reckless lunge on the hour - but Aidy Boothroyd's side held on for a draw.
Managerless Coventry, who sacked Aidy Boothroyd a fortnight ago, are 20th in the Championship - just seven points away from danger.
Former Wales captain Gary Speed has been approached by Swansea while ex-Watford boss Aidy Boothroyd and Exeter manager Paul Tisdale have been linked.
Poyet has already held talks with Swansea chairman Huw Jenkins, as has Sheffield United player coach Gary Speed and former Watford manager Aidy Boothroyd.
Former Watford boss Aidy Boothroyd is also known to be keeping an interest in the vacancy as Jenkins confirmed the 38-year-old had "declared an interest".
Watford boss Aidy Boothroyd has earned a reputation for motivating his players - and it is clear that installing confidence in his team is one of his fortes.
Another Premiership manager who has seen Healy at close quarters is Watford's Aidy Boothroyd, who was on the staff at Leeds when the striker moved to Elland Road from Deepdale.
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Former Wales captain Gary Speed, ex-Watford manager Aidy Boothroyd and Exeter City's back-to-back promotion winning boss Paul Tidsdale have been linked with the Swans job that Martinez helped make so appealing.
Watford boss Aidy Boothroyd waited for the early stages of the second half to unfold before making a bold change, sending on striker Tamas Priskin for defender Adrian Mariappa after 57 minutes.
Former Premier League referee Graham Poll said the decision was inexplicable and had sympathy for the Watford players and manager, Aidy Boothroyd, who was sent to the stands after remonstrating with the officials.
After surviving early pressure from Swansea, whose best chance fell to a Craig Beattie snap-shot, boss Aidy Boothroyd's Coventry took control, but they could not find a way past the excellent Dorus de Vries.
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