In the literature of discrete-valued time series modelling, various bivariate integer-valued autoregressive time series models of order 1 (BINAR(1)) have been proposed particularly based on the binomial thinning mechanism and with different innovation distributions. These BINAR(1)s are mostly suitable for modelling bivariate counting series of varied levels of overdispersion. Recently, in the context of overdispersion, the INAR(1) with Poisson-Lindley (PL) innovations have been introduced to provide superior model fitness criteria than other competing INAR(1)s. Thus, this paper proposes to develop classes of BINAR(1)PL, including BINAR(1)PL(I) and BINAR(1)PL(II), under different cross-correlation functions. The parameter estimations are conducted via the conditional maximum likelihood (CML) approach. Monte Carlo simulation experiments are implemented to assess the asymptotic properties of the CML estimators under different combinations of the cross-correlation parameters. The proposed models are also applied to the Pittsburg crimes data and compared with other competing popular overdispersed BINAR(1) models.