Jonas Where chastity is a sex game or fetish then no, parents can't really be involved, but where it's a pure personal development thing, help to enforce rules that you want to follow, parental authority is about the safest form of external control you could possibly find.
I put the best spin I could think of on parental involvement above. The question seems to hinge on whether those involved, and those viewing, consider chastity to always be a sexual activity.
If chastity is considered sexual, then "Sex with parents wtf?! Eeew, gross!". However, where it's viewed as non-sexual then chastity could be viewed as a simple agreement between adults.
Logically, chastity should be viewed as non-sexual endeavour, if an honest priest tells me to remain chaste he's not having sex with me. However, the world view is currently being set by the kink/fetish crowd who (as @MissBlossom quotes) see chastity as the ultimate sexual perversion and project that view onto all.
Not every situation is a scene. However, yeah, in general, I too am very sceptical of any and all parental involved reported on this forum.