We investigated the possibility of estimating network link weights from the multivariate time series of phase oscillators on a complex network. The inverse phase synchronization index of the coupled oscillator network is found to grow in proportion to the corresponding link weight, as network synchronization occurs for a strong coupling strength. This implies that the network link weights can be estimated from the measurement of the inverse phase synchronization indices. By adopting this estimation method, we successfully reconstructed the minimal spanning tree of the original network from the inverse phase synchronization indices. Even for the weak coupling case, the estimation of the network link weights could be improved significantly by taking the average of a sufficiently large number of configurations.