Unless you ask for minimal Fano, this is false. Take a rational curve $C$ on a Fano manifold $M$ (say, del Pezzo surface), and consider a family of blow-ups of $M$ parametrized by $C$, obtained by blowing up a point in $C$. A blow-up is often Fano, but this family is not always isotrivial (say, for an appropriate choice of del Pezzo $M$).

Unless you ask for minimal Fano, this is false. Take a rational curve $C$ on a Fano manifold $M$ (say, del Pezzo surface), and consider a family of blow-ups of $M$ parametrized by $C$, obtained by blowing up a point in $C$. A blow-up is often Fano, but this family is not always isotrivial (say, for an appropriate choice of del Pezzo $M$).

Update: for del Pezzo this argument does not work - sorry!