We experimentally demonstrate spatial beam self-cleaning and supercontinuum generation in a tapered Ytterbium-doped multimode optical fiber with parabolic core refractive index and doping profile when 1064 nm pulsed beams propagate from wider (120 micrometers) into smaller (40 micrometers) diameter. In the passive mode, increasing the input beam peak power above 20 kW leads to a bell-shaped output beam profile. In the active configuration, gain from the pump laser diode permits to combine beam self-cleaning with supercontinuum generation between 520-2600 nm. By taper cut-back, we observed that the dissipative landscape i.e., a non-monotonic variation of the average beam power along the MMF leads to modal transitions of self-cleaned beams along the taper length.