This paper considers the problem of estimating parameters of the bivariate exponential distribution with a location parameter for a two-component shared parallel system using component data from system-level life test terminated at the time of the prespecified number of system failure. In the system-level life testing, there are three patterns of failure types ; 1) both component failed 2) both component censored 3) one is failed and the other is censored. In the third case, we assume that the failure time might be known or unknown. The maximum likelihood estimators are obtained for the case of known/unknown failure time when the other component is censored.