We investigate the condensation phase transitions of conserved-mass aggregation (CA) model on weighted scale-free networks (WSFNs). In WSFNs, the weight w_ij is assigned to the link between the nodes i and j. We consider the symmetric weight given as w_ij=(k_i k_j)^α. In CA model, the mass m_i on the randomly chosen node i diffuses to a linked neighbor of i,j, with the rate T_ji or an unit mass chips off from the node i to j with the rate ω T_ji. The hopping probability T_ji is given as T_ji=

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... ji/∑_ w_li, where the sum runs over the linked neighbors of the node i. On the WSFNs, we numerically show that a certain critical α_c exists below which CA model undergoes the same type of the condensation transitions as those of CA model on regular lattices. However for α≥α_c, the condensation always occurs for any density ρ and ω. We analytically find α_c = (γ-3)/2 on the WSFN with the degree exponent γ. To obtain α_c, we analytically derive the scaling behavior of the stationary distribution P^∞_k of finding a walker at nodes with degree k, and the probability D(k) of finding two walkers simultaneously at the same node with degree k. We find P^∞_k ∼ k^α+1-γ and D(k) ∼ k^2(α+1)-γ respectively. With P^∞_k, we also show analytically and numerically that the average mass m(k) on a node with degree k scales as k^α+1 without any jumps at the maximal degree of the network for any ρ as in the SFNs with α=0.