\textit{L. C. Hsu} and \textit{P. J. S. Shiue} [Adv. Appl. Math. 20, No. 3, 366--384 (1998; Zbl 0913.05006)] introduced a far-reaching generalization of Stirling numbers, \(S(n,k;\alpha,\beta,r)\), and they gave eleven known combinatorial sequences as specializations. \textit{B. Bényi} et al. [Integers 22, Paper A79, 28 p. (2022; Zbl 1501.11040)] provided ten further interesting specializations of \(S(n,k;\alpha,\beta,r)\) . The paper under review somewhat extends the range when \(S(n,k;\alpha,\beta,r)\) is defined and denotes it by \(\Big\{ \frac{n}{k}\Big\}_{(\alpha,\beta,r)}\), shows that two kinds of central factorial numbers (see [\textit{P. L. Butzer} et al., Numer. Funct. Anal. Optim. 10, No. 5--6, 419--488 (1989; Zbl 0659.10012)]) also come from these generalized Stirling numbers as \(t(n,k)= \Big\{ \frac{n-1}{k-1}\Big\}_{(1,0,\frac{n}{2}-1)}\) and \(T(n,k)=\Big\{ \frac{n}{k}\Big\}_{(0,1,-\frac{k}{2})}\). The paper actually investigates a new family of numbers, \textit{balanced Stirling numbers}, shows that balanced Stirling numbers are among the \(S(n,k;\alpha,\beta,r)\) numbers, and that the central factorial numbers are specializations of the balanced Stirling numbers.