  We first study the drift parameter estimation of the fractional
Ornstein-Uhlenbeck process (fOU) with periodic mean for every
$\frac{1}{2}<H<1$. More precisely, we extend the consistency proved in
\cite{DFW} for $\frac{1}{2}<H<\frac{3}{4}$ to the strong consistency for any
$\frac{1}{2}<H<1$ on the one hand, and on the other, we also discuss the
asymptotic normality given in \cite{DFW}. In the second main part of the paper,
we study the strong consistency and the asymptotic normality of the fOU of the
second kind with periodic mean for any $\frac{1}{2}<H<1$.
