Pet resists bouncing around relative (10)
I believe the answer is:
stepsister
'relative' is the definition.
(stepsister is a kind of relative)
'pet resists bouncing' is the wordplay.
'bouncing' indicates anagramming the letters.
'pet'+'resists'='petresists'
'petresists' anagrammed gives 'STEPSISTER'.
'around' acts as a link.
(Other definitions for stepsister that I've seen before include "Relation gained through a second marriage" , "one who hated Cinderella" , "relative in law" , "member of the family" .)