Central limit theorems and asymptotic properties of the minimum-contrast estimators of the drift parameter in linear stochastic evolution equations driven by fractional Brownian motion are studied. Both singular ($H < \frac{1}{2})$ and regular ($H > \frac{1}{2})$ types of fractional Brownian motion are considered. Strong consistency is achieved by ergodicity of the stationary solution. The fundamental tool for the limit theorems and asymptotic normality (shown for Hurst parameter $H < \frac{3}{4}$) is the so-called $4^{th}$ moment theorem considered on the second Wiener chaos. This technique provides also the Berry-Esseen-type bounds for the speed of the convergence. The general results are illustrated for parabolic equations with distributed and pointwise fractional noises.