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Understanding how thermal-phonon paths can be shaped is key for controlling heat dissipation at the nanoscale. Thermophononic crystals are periodic porous nanostructures with thermal conductivity deviating from effective medium theory, which is possible if the characteristic sizes are of the order of phonon mean free paths and/or if phonons are forced to flow in privileged directions. We investigate suspended silicon nanomembranes with a periodic array of partially perforated holes of original paraboloid shape, with all characteristic lengths below 100 nm. Results from scanning thermal microscopy, a thermal sensing technique derived from atomic force microscopy, indicate that partial perforation of the membranes impacts heat conduction moderately, with the holey crystals showing a thermal conductivity reduction by a factor 6 in comparison to the bulk and a factor 2.5 in comparison to the non-perforated membrane. The impact of the phononic shapes is analyzed in light of a complementary Monte Carlo ray-tracing estimate of the effective phonon mean free paths that include multiple phonon reflection and highlights phonon backscattering.