First, both parabolic & Yagi antennas are "directional" in that their main lobe is designed to have high gain in a limited direction. this is because all antennas have to trade the width of their main beam(s) with the amount of gain in that beam. you can either put a little power everywhere or a lot of power in one direction.
that being said the picture you show is also from wikipedia. From that Yagi antenna article
The Yagi was first widely used during World War II for airborne radar sets, because of its simplicity and directionality
the subtext on the photo reads:
A Nakajima J1N1-S night fighter with quadruple Yagi radar transceiver antennas
Reading through the J1N article it appears these were used for night surveillance and intercept. Much like modern fighters with radars antennas in the front, the likely use was for tracking or targeting targets - probably B29's. With the Yagi (or any highly directive design) the radar can 'see farther' since it has high gain in the forward direction.
With 4 antennas with their beams pointed in slightly different directions their coverage would have been a bit wider (both in azimuth and elevation) for searching.