Admittance of the circuit:

$Y= \bar{Y}_1 + \bar{Y}_2 \\ = \dfrac{R-jX_L}{R^2+ X_L^2}+ \dfrac{j}{X_C} \\ = \dfrac{R}{R^2+ X_L^2} -j \bigg( \dfrac {X_L}{R^2+ X_L^2}- \dfrac{1}{X_C} \bigg)$

At resonance,

$\dfrac{X_L}{R^2+ X_L^2}- \dfrac{1}{X_C} =0 \\ \dfrac{X_L}{R^2+ X_L^2}= \dfrac{1}{X_C} \\ X_L X_C= R^2+ X_L^2 \\ ω_0 L.\dfrac{1}{ω_o C}= R^2+ ω_0^2 L^2 \\ ω_0^2 L^2= \dfrac LC .R^2 \\ ω_0^2= \dfrac{1}{LC}-\dfrac{R^2}{L^2} \\ \boxed{f_0= \dfrac{1}{2π} \sqrt{\dfrac{1}{LC}-\dfrac{R^2}{L^2}}}$