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It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric family of number fields. In this paper we consider the infinite parametric family of simplest quartic fields $K$ generated by a root $\xi$ of the polynomial $P_t\left(x\right)=x^4-t{x}^3-6x^2+tx+1$, assuming that $t>0$, $t\ne 3$ and $t^2+16$ has no odd square factors. In addition to generators of power integral bases we also calculate the minimal...