This paper studies the behaviour of quadratic variations of a stochastic wave equation driven by a noise that is white in space and fractional in time. Complementing the analysis of quadratic variations in the space component carried out in [Correlation structure, quadratic variations and parameter estimation for the solution to the wave equation with fractional noise, Electron. J. Stat. 12 (2018), no. 2, 3639–3672] and [Generalized k k -variations and Hurst parameter estimation for the fractional wave equation via Malliavin calculus, J. Statist. Plann. Inference 207 (2020), 155–180], it focuses on the time component of the solution process. For different values of the Hurst parameter a central and a noncentral limit theorems are proved, allowing to construct parameter estimators and compare them to the findings in the space-dependent case. Finally, rectangular quadratic variations in the white noise case are studied and a central limit theorem is demonstrated.