Parent-of-origin effects, also known as genomic imprinting, exist for many mammalian genes. For imprinted genes the expression of an allele depends upon the sex of the transmitting parent. Here we have developed a method based on alleles that are shared identical by descent by affected sib pairs, that allows for parent-of-origin effects. Our method allows for sex-specific recombination rates, an important consideration in studying imprinted genes. We have also derived a tetrahedron for the true identical-by-descent frequencies accounting for parent-of-origin effects. Using this tetrahedron, we propose a robust generalized minmax test for linkage and discuss its properties in the presence of genomic imprinting. We have also performed power comparisons of various allele sharing tests and provide regions of the tetrahedron in which the different tests are optimal. We also provide useful strategies to determine the optimal tests to use while performing a genome scan.