The truncated matrix Stieltjes moment problem consists in the following: given a sequence \(S_0\), \(S_1\), \(\dots\), \(S_m\) of \(p\times p\) Hermitian matrices, find a positive measure \(d\sigma(x)\) supported by \([0,\infty)\) such that \[ S_k=\int_0^\infty x^k \,d\sigma(x), \quad k=0,1,\dots,m.\tag{1} \] In the scalar case a unified approach both to the even \((m=2n)\) and to the odd \((m=2n-1)\) case was proposed by \textit{M.G.~Kreĭn} [Am. Math. Soc., Transl., II. Ser. 95, 219--234 (1970); translation from Mat. Issled. 2, No. 2, 114--131 (1967; Zbl 0219.44010)]. In the matrix case this problem was investigated by \textit{Yu. Dyukarev} [The matrix Stieltjes moment problem, Manuscript deposited at VINITI, No. 2628, 22.03.1981, Moscow (H) 1981] and \textit{V. Bolotnikov} [Z. Anal. Anwendungen 14 (3), 441--468 (1995; Zbl 0828.47012)] using the Potapov's method in the nondegenerate and degenerate cases, respectively. In the present paper the authors apply the step-by-step process of solving the degenerate moment problem \((1)\) in the indeterminate case. A parametrization of all the solutions of the problem \((1)\) is given both for the even and odd cases. This approach leads to a natural factorization of the corresponding resolvent matrix.