Maybe. The parallel is certainly there, but based on the evolution of ascetic/monastic practice, it looks more like women began creating parallel structures to the monastic communities of Christian men more than the ritual community of the Vestal Virgins.

We need to distinguish between female asceticism (living a life of self-denial) from female monasticism (living in a religious community governed by rules that may often be considered ascetic).

Women's asceticism initially begins with women practicing asceticism in the home, often as widows or young women who refuse marriage. Communities of these women that parallel the institution of the Vestal Virgins don't really start up until after men are building these communities. The reason probably has more than a little to do with Roman attitudes towards the social role of women, who they expect to be under the authority of a male family member. By staying in the home, they're not stressing out the neighbors.

If you read early church documents such as the Life of Antony you see that many early Christians grounded that initial aestheticism in Matthew 19:21 "If you would be perfect, go, sell your possessions and give the money to the poor" (NRSV).

Female monasticism on the other hand seems to have derived from a series of evolutions in early Christianity that started with men attempting an ascetic life, then gathering in communities for mutual support and regulation, then slowly and begrudgingly opening their lifestyle up to women, who basically kicked the door down and demanded to be let in.