Wilf [Accelerated series for universal constants, by the WZ method, Discrete Math. Theor. Comput. Sci. 3 (1999) 189-192] applied Zeilberger's algorithm to obtain an accelerated version of the famous series Pk1=1 1=k2 = 2=6. However, if we write the Basel series P1 k=1 1=k2 as a 3F2(1)-series, it is not obvious as to how to determine a Wilf-Zeilberger (WZ) pair or a WZ proof certificate that may be used to formulate a proof for evaluating this 3F2(1)-expression. In this article, using the WZ method, we prove a remarkable identity for a 3F2(1)-series with three free parameters that Maple 2020 is not able to evaluate directly, and we apply our WZ proof of this identity to obtain a new proof of the famous formula (2) = 2=6. By applying partial derivative operators to our WZ-derived 3F2(1)-identity, we obtain an identity involving binomial-harmonic sums that were recently considered by Wang and Chu [Series with harmonic-like numbers and squared binomial coefficients, Rocky Mountain J. Math., In press], and we succeed in solving some open problems given by Wang and Chu on series involving harmonic-type numbers and squared binomial coefficients.