Understanding thermal transport through nanoscale van der Waals interfaces is vital for addressing thermal management challenges in nanoelectronic devices. In this work, the interfacial thermal conductance (GCA) between copper phthalocyanine (CuPc) nanoribbons is reported to be on the order of 10^5 Wm-2K-1 at 300 K, which is over two orders of magnitude lower than the value predicted by molecular dynamics (MD) simulations for a perfectly smooth interface between two parallelly aligned CuPc nanoribbons. Further MD simulations and contact mechanics analysis reveal that surface roughness can significantly reduce the adhesion energy and effective contact area between CuPc nanoribbons, and thus result in an ultralow GCA. In addition, the adhesion energy at the interface also depends on the stacking configuration of two CuPc nanoribbons, which may also contribute to the observed ultralow GCA.