Traditionally, it is assumed that the lead time spare part demand follows the Poisson or normal distribution. It is well-known that the mean of Poisson distribution equals to its variance, which is called Poisson dispersion. Actual demand data scarcely meet the Poisson dispersion. To overcome this problem, several two-parameter variants (e.g., zero-inflated Poisson and hurdle shifted Poisson) of the Poisson distribution have been used for modelling irregular spare part demand data. In this paper, we propose two new two-parameter alternatives, which are transmuted and exponentiated Poisson distributions. The proposed distributions can be over-or under-Poisson dispersive, and hence provide better flexibility without losing the simplicity of the Poisson distribution too much. The appropriateness of the proposed models is illustrated through three real-world examples. Based on the idea that a fitted distribution should provide a good goodness-of-fit to the right tail of the empirical demand distribution so as to achieve good inventory performance, a weighted least square method is also proposed to estimate the model parameters. The appropriateness of the proposed approach is also illustrated.