Clingan put Coventry ahead seven minutes before the break when he sent an unstoppable free-kick into the top corner after Paul Gallacher had handled on the left-hand side of the box.
It was his third goal in his last four Championship matches and looked for a while to have set the Sky Blues on their way to victory.
But Waghorn, on loan from Sunderland, earned Leicester a point 19 minutes from time when he combined with fellow substitute Steve Howard and beat Keiren Westwood with a shot on the run from the edge of the box.
Coventry boss Chris Coleman made two changes to the side that came from behind to beat Watford with Martin Cranie and Freddy Eastwood replacing Jordan Clarke, who dropped down to the bench, and Michael McIndoe respectively.
Leicester were unchanged following their first away league win at Middlesbrough but boss Nigel Pearson included striker DJ Campbell in his squad for the first time this season.
Chris Weale was the first goalkeeper called into action after 10 minutes to tip Aron Gunnarsson's shot round a post.
At the other end, opposite number Westwood was a mere bystander as Gallacher's curling shot struck the crossbar in the 21st minute. Gallacher then handled to allow Clingan to score.
Weale tipped another Gunnarsson effort around a post early in the second half and Elliott Ward sent a shot just wide soon afterwards as Coventry looked to put daylight between themselves and their visitors.
Pearson made a triple substitution on the hour mark, bringing on Howard, Campbell and Waghorn, and 11 minutes later it paid dividends as the Foxes equalised.
Both sides pushed for a winning goal but neither created a clear-cut scoring opportunity.
Championship player of the month for September Leon Best and his strike partner Clinton Morrison had shots blocked while defender Jack Hobbs headed wide for Leicester.