Getting the power of a signal from its Fourier transform?
Note that due to the periodic component in the signal ($A\cos(\omega_1t))$, the integral
$$\int_{-\infty}^{\infty}|x(t)|^2dt=\int_{-\infty}^{\infty}|X(f)|^2df\tag{1}$$
doesn't exist, i.e., the energy of the given signal is infinite. The signal is a so-called power signal, having finite power and infinite energy.
What you can do is compute its power using the equation given in your question:
$$P_x=\lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}|x(t)|^2dt\tag{2}$$
but you can't use Parseval's theorem to compute that limit.
However, you can use Parseval's theorem indirectly to show that the contributions to the limit $(2)$ from all components of $|x(t)|^2$ containing the $\textrm{sinc}$ function go to zero as $T\to\infty$, because all those components have finite energy, hence zero power. So the power of the given signal just equals the power of $A\cos(\omega_1t)$, and I'm sure you know how to compute that power.