The purpose of this paper is to develop a reduced Crouzeix-Raviart immersed finite element method (RCRIFEM) for two-dimensional elasticity problems with interface, which is based on the Kouhia-Stenberg finite element method (Kouhia et al. 1995) and Crouzeix-Raviart IFEM (CRIFEM) (Kwak et al. 2017). We use a P-1-conforming like element for one of the components of the displacement vector, and a P-1-nonconforming like element for the other component. The number of degrees of freedom of our scheme is reduced to two thirds of CRIFEM. Furthermore, we can choose penalty parameters independent of the Poisson ratio. One of the penalty parameters depends on Lame's second constant mu, and the other penalty parameter is independent of both mu and lambda. We prove the optimal order error estimates in piecewise H-1-norm, which is independent of the Poisson ratio. Numerical experiments show optimal order of convergence both in L-2 and piecewise H-1-norms for all problems including nearly incompressible cases.