Weak interchain interactions have been considered to be a bottleneck for heat transfer in polymers, while covalent bonds are believed to give a high thermal conductivity to polymer chains. For this reason, cross-linkers have been explored as a means to enhance polymer thermal conductivity; however, results have been inconsistent. Some studies show an enhancement in the thermal conductivity for polymers upon cross-linking, while others show the opposite trend. In this work we study the mechanisms of heat transfer in cross-linked polymers in order to understand the reasons for these discrepancies, in particular examining the relative contributions of covalent (referred to here as "bonding") and nonbonding (e.g., van der Waals and, electrostatic) interactions. Our results indicate cross-linkers enhance thermal conductivity primarily when they are short in length and thereby bring polymer chains closer to each other, leading to increased interchain heat transfer by enhanced nonbonding interactions between the chains (nonbonding, interactions being highly dependent on interchain distance). This suggests that enhanced nonbonding interactions, rather than thermal pathways through cross-linker covalent bonds, are the primary transport mechanism by which thermal conductivity is increased. We further illustrate this by showing that energy from THz acoustic waves travels significantly faster in polymers when nonbonding interactions are included versus when only covalent interactions are present. These results help to explain prior studies that measure differing trends in thermal conductivity for polymers upon cross-linking with various species.