The videos on the previous page showed conceptually how the calculation of forces in a supercell in which an atom is displaced, can lead to the full phonon spectrum. To do this in practice you need (1) a DFT code that can calculate forces in supercells (something every mainstream DFT code can do), and (2) an auxiliary code that generates the required supercells with displaced atoms, that collects the calculated forces, and uses them to determine the phonon spectrum. That auxiliary code should be able to communicate with the DFT code.
A free and open source code that is very much used nowadays for phonon calculations, is Phonopy. It can communicate with many modern DFT codes. Another example (not free) is Phonon (its website gives some short and good basic phonon information).
Rita Omamuyovwi Akemu, a former student of this online course, wrote a step by step guide about how to use Phonopy together with Quantum Espresso for an example crystal. Jan Jaeken, a phonon expert working then at Ghent University, supervised this effort. You find the step-by-step guide here.