The present study initially identifies the generalized symmetric connections of type $(\alpha,\beta)$, which can be regarded as more generalized forms of quarter and semi-symmetric connections. The quarter and semi-symmetric connections are obtained when $(\alpha,\beta)=(1,0)$ and $(\alpha,\beta)=(0,1)$, respectively. Taking this into account, a new generalized symmetric metric connection is attained on Lorentzian para-Sasakian manifolds. In accordance with this connection, some results are obtained through calculation of tensors belonging to a Lorentzian para-Sasakian manifold involving the curvature tensor, the Ricci tensor and the conformal curvature tensor.