Summary: A problem of drift parameter estimation is studied for a nonergodic weighted fractional Vasicek model defined as \(d{X_t}=\theta (\mu +{X_t})dt+d{B_t^{a,b}}, t\ge 0\), with unknown parameters \(\theta >0\), \(\mu \in \mathbb{R}\) and \(\alpha :=\theta \mu\), whereas \({B^{a,b}}:=\{{B_t^{a,b}},t\ge 0\}\) is a weighted fractional Brownian motion with parameters \(a>-1, |b|<1, |b|<a+1\). Least square-type estimators \((\widetilde{\theta}_T, \widetilde{\mu}_T)\) and \((\widetilde{\theta}_T,\widetilde{\alpha}_T)\) are provided, respectively, for \((\theta, \mu)\) and \((\theta ,\alpha)\) based on a continuous-time observation of \(\{{X_t},\hspace{2.5pt}t\in [0,T]\}\) as \(T\to \infty\). The strong consistency and the joint asymptotic distribution of \((\widetilde{\theta}_T, \widetilde{\mu}_T)\) and \((\widetilde{\theta}_T, \widetilde{\alpha}_T)\) are studied. Moreover, it is obtained that the limit distribution of \(\widetilde{\theta}_T\) is a Cauchy-type distribution, and \(\widetilde{\mu}_T\) and \(\widetilde{\alpha}_T\) are asymptotically normal.