We study the phase behavior of two lattice polymer models (self-avoiding walks) incorporating attractive short-range interactions between parallel chain sections, attempting to mimick hydrogen bonding between monomers. The investigation is carried out in the framework of the Bethe approximation on a hypercubic lattice. The former model, which includes attraction between parallel sections longer than one chain segment, has been recently studied in full detail in the two-dimensional case, by means of accurate numerical transfer matrix techniques, and displays a first-order transition from a swollen ("coil") state to a collapsed ordered ("solid") state. Our investigation on this model is mainly meant to show that the Bethe approximation is able to recover such a behavior, and to extend the result to three dimensions. The latter model is similar, but takes into account attractive interactions also between one segment long parallel sections, which have no reason to be neglected in principle. We obtain, both in two and three dimensions, two phase transitions: An ordinary theta-collapse from the coil state to an isotropic compact ("globule") state, and a first-order transition from the globule to the solid phase. Such results are compared with those of previously investigated polymer models, and their relevance to describe the formation of protein secondary structure is also briefly discussed.