Finite geometry (FG) codes combine the algebraic properties of classical block codes with the iterative belief propagation (BP) decoding ability of low-density parity-check~(LDPC) codes. However, exploiting both advantages in practice is hindered by the fact that the standard incidence matrix between $(μ+1)$-flats and points is dense and contains many short cycles for any flat dimension $μ\geq 1$. In this work, we propose to sparsify the decoding matrix based on pencil selection, formulated as a constant-dimension subspace packing problem and solved explicitly using lifted Gabidulin codes. For both affine and projective geometries, sparse parity-check matrices are constructed and verified for FG codes of lengths up to $1024$. Simulations on four FG codes show no visible error floor and around $0.5$~dB gain over corresponding 5G LDPC codes at a block error rate of $10^{-7}$.