A class of efficient parallel multivalue hybrid methods for stiff differential equations are presented,which are all extremely stable at infinity, A-stable for orders 1-3 and A(α)-stable for orders 4-8. Eachmethod of the class can be performed parallelly using two processors with each processor having almost thesame computational amount per integration step as a backward differentiation formula (BDF) of the same orderwith the same stepsize performed in serial, whereas the former has not only much better stability propertiesbut also a computational accuracy higher than the corresponding BDF. Theoretical analysis and numericalexperiments show that the methods constructed in this paper are superior in many respects not only to BDFsbut also to some other commonly used methods.