In this paper, we consider estimation of unknown parameters of the tapered Pareto distribution, which belongs to the class of semiheavy distributions, by a sample with excluded l_n largest and k_n smallest observations. We establish necessary and sufficient conditions in terms of proportions k_n/n and l_n/n for weak consistency and joint asymptotic normality of parameterizedmoment-type estimators for the shape and form parameters. Additionally, we extend the result on weak consistency of generalized Hill statistics (introduced in [V. Paulauskas and M. Vaiˇciulis, On the improvement of Hill and some others estimators, Lith. Math. J., 53(3):336–355, 2013]) to the case where the extreme value index is not positive. We demonstrate the performance of the proposed estimators on both simulated data from the tapered Pareto distribution and real data from finance.