Three-parameter (3-p) Weibull distribution is widely used to model failure distribution in reliability studies, and so the estimation of the parameters of this distribution is very crucial. The maximum likelihood (ML) method is the most popular method among parameter estimation methods. However, likelihood equations do not have explicit solutions for the 3-p Weibull distribution. Therefore, using metaheuristic methods is logical to obtain the ML estimation of 3-p Weibull distribution. The artificial bee colony (ABC) is one of the very simple, robust and population-based stochastic metaheuristic algorithms. The aim of this study is to obtain ML estimations of parameters of 3-p Weibull distribution by suggesting ABC with Levy flights (LABC) which improve the exploitation ability of the ABC algorithm. Furthermore, the ML estimation results of the LABC algorithm are compared with other well-known metaheuristic algorithms, standard ABC, particle swarm optimization, particle swarm optimization with Levy flights, simulated annealing, differential evolution and genetic algorithm through simulation studies and a real-data application. The results show that the suggested LABC algorithm gives more accurate ML estimations than the other metaheuristic algorithms for the parameter estimation of 3-p Weibull distribution.