We study the problem of unconditional uniqueness and non-uniquenesss for Hardy-Hénon parabolic equations in weighted Lorentz space. We mainly focus on the Fujita case to see that the second exponent of the Lorentz space plays a crucial role in determining the sharp threshold for uniqueness. This talk is based on the joint work with M. Ikeda (RIKEN/Keio), K. Taniguchi (Tohoku) and S. Tayachi (Univ. of Tunis El Manar) (arXiv:2301.00506).