In this story, Yukio is actually...wait for it...a human! His father died just after he was born and his mother is Yuri. Rin is born when Yukio is eight. Yukio was raised by Yuri and her old friend Shiro Fujimoto. Yuri dies in childbirth and Shiro takes the boys in as his own sons. Yukio promises that he'll protect Rin, no matter what. Yukio becomes an exorcist to fulfill that promise and Rin grows up not knowing who his father is or about his flames.
Everything is perfectly fine until Rin is seven. Then, demons start going after him. Shiro has to take drastic measures to keep him safe and eventually, Yukio must step in too. Satan never learned of Rin's existence, but his half brother, Iblis, has and is more than ready to absorb Rin's flames(his soul) to be more powerful than all the other Demon Kings.