No, you have to be the one to sponsor him, your parents can't do it.
But don't worry, in order to sponsor only a common-law partner, the sponsor doesn't have an income requirement he or she should meet. The only requirement is that you (the sponsor) are not bankrupt or on social assistance. And, if you like, you can also include a letter from your parents saying they'll help you out for a while while you both get settled in Canada.
I suggest you look over the forms and guides, even if you're not ready to apply yet. Just to give you an idea of everything that you will need.
http://www.cic.gc.ca/english/information/applications/fc.aspIf you intend to sponsor him while still living in the UK, you will need to include evidence to show that you intend to move to Canada when he is granted his permanent residency. People do that by showing they're selling a house abroad or including a letter from the landlord that they know of their plans to move soon, providing job offers or school enrollment in Canada, letters from family and friends who know of your plans to move or are prepared to help you while you do so,...