Here we present a theoretical study of the thermoelectric transport through [2,2]paracyclophane-based single-molecule junctions. Combining electronic and vibrational structures, obtained from density functional theory (DFT), with nonequilibrium Green's function techniques allows us to treat both electronic and phononic transport properties at a first-principles level. For the electronic part, we include an approximate self-energy correction, based on the DFT+$\Sigma$ approach. This enables us to make a reliable prediction of all linear response transport coefficients entering the thermoelectric figure of merit $ZT$. Paracyclophane derivatives offer a great flexibility in tuning their chemical properties by attaching different functional groups. We show that, for the specific molecule, the functional groups mainly influence the thermopower, allowing us to tune its sign and absolute value. We predict that the functionalization of the bare paracyclophane leads to a largely enhanced electronic contribution $Z_{\mathrm{el}}T$ to the figure of merit. Nevertheless, the high phononic contribution to the thermal conductance strongly suppresses $ZT$. Our work demonstrates the importance to include the phonon thermal conductance for any realistic estimate of the $ZT$ for off-resonant molecular transport junctions. In addition, it shows the possibility of a chemical tuning of the thermoelectric properties for a series of available molecules, leading to equally performing hole- and electron-conducting junctions based on the same molecular framework.