A family of lifetime distributions is considered which covers many distributions as its special cases. Two measures of reliability are studied, R(t)=P(X > t) and P = P(X > Y). Assuming the availability of some prior information on the parameter of interest, shrinkage estimators are developed for the powers of the scale parameter and reliability functions based on records. These proposed estimators are compared with the maximum likelihood estimators and uniformly minimum variance unbiased estimators of the parametric functions in terms of their relative efficiency. We establish that all the proposed estimators outperform the usual estimators as the true value of parameters approaches their hypothesized value.