Mazirek's last ability is a triggered ability, not an activated ability. It doesn't allow you to sacrifice a permanent whenever you want; rather, you need some other way of sacrificing permanents.

If you sacrifice a permanent as part of casting a spell or activating an ability, Mazirek's last ability will resolve before that spell or ability.

A legendary permanent that is put into a graveyard because of the "legend rule" isn't sacrificed.

If you sacrifice Mazirek and other permanents at the same time, Mazirek's last ability triggers for the other permanents.