$y = \frac{a}{2}$

$ \Rightarrow \frac{a}{2} = a\sin (\omega \,t + {\phi _0})$

$ \Rightarrow (\omega \,t + {\phi _0}) = \phi = \frac{\pi }{6}$ or $\frac{{5\pi }}{6}$
Physical meaning of $\phi = \frac{\pi }{6}$ : Particle is at point $P$ and it is going towards B
Physical meaning of $\phi = \frac{{5\pi }}{6}$ : Particle is at point $P$ and it is going towards $O$
So phase difference $\Delta \phi = \frac{{5\pi }}{6} - \frac{\pi }{6} = \frac{{2\pi }}{3} = 120^\circ $