Given an undirected connected graph \(G = (V, E, \rho , \mathcal {X}, \mathcal {Y})\) , where \(\rho : E \rightarrow \textrm{R}^{+} \cup \{ 0 \}\) is an edge-weight function, \(\mathcal {X} \subset V\) is a subset of clients, and \(\mathcal {Y} \subset V\) is a subset of candidates, and a positive integer \(k < |\mathcal {Y}|\) , the k-Supplier Problem (k SP) asks for an optimal subset of \(\mathcal {Y}\) of cardinality at most k in G to minimize the radius from \(\mathcal {X}\) to the subset. In this paper, we focus on the case of \(\mathcal {X} \cap \mathcal {Y} \ne \emptyset , \mathcal {X}, \mathcal {Y}\) , and consider the scenario where the shortest path distances \(d(\cdot , \cdot )\) satisfy a parameterized triangle inequality between \(\mathcal {X}\) and \(\mathcal {Y}\) , i.e., \(d(v, w) + d(w, u) \ge \alpha \cdot d(v, u)\) , \(1 < \alpha \le 2\) , for any \(v, u \in \mathcal {X}, v \ne u\) and \(w \in \mathcal {Y}\) . We present a two-stage dual approximation algorithm KSP-PG for the kSP with parameterized triangle inequality between \(\mathcal {X}\) and \(\mathcal {Y}\) . If KSP-PG stops at the end of Stage 1 then it obtains a \(\frac{2}{\alpha }\) -approximation, and if it stops at the end of Stage 2 then it obtains a \((\frac{2}{\alpha ^{2}} + \frac{1}{\alpha })\) -approximation. It is implied by \(\frac{2}{\alpha } \in [ 1, 2 )\) and \(\frac{2}{\alpha ^{2}} + \frac{1}{\alpha } \in [ 1, 3 )\) that KSP-PG is better than the previously best 3-approximation algorithm of Hochbaum and Shmoys (J. ACM. 33: 533–550, 1986). Regardless of the parameterized aspect, KSP-PG obtains a 2-approximation if it stops at the end of Stage 1 and a 3-approximation if it stops at the end of Stage 2, for the kSP.