Sage Bhringi was an ardent devotee of Lord Shiva, but he never considered Parvati to be a part of his worship.
As always, sage Bhringi meditated over Lord Shiva at his abode at Mount Kailasa. At the end of his worship he would religiously circum-ambulate Lord Shiva. Parvati having seen this ritual of Bhringi, chose to be part of it and sat closer to Lord Shiva. Sage Bhringi noticed this, and realizing what had just happened, turned himself into a bee and went around Shiva thrice.
Parvati was enraged and Shiva wanted to teach Bhringi a lesson that Parvati is an integral part of Him and cannot be excluded. He merged Himself with Parvati such that Bhringi would have to circum-ambulate both of them.
Bhringi is often depicted as a skeleton in most images, this is because Parvati cursed him in fury to become thin and completely emaciated, so weak that he could not stand. Shiva, moved by Bhringi’s plight, gave a 3rd leg for support.
Hence Bhringi is always depicted with three legs
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