This work presents a combined methodology—referred to as SSPINNs—that integrates optimal sensor placement via sparse sensing (SS) with physics-informed neural networks (PINNs) to reconstruct temperature and velocity fields in axisymmetric flames. The SS procedure determines optimal positions for a limited number of point measurements taken in soot-free regions (e.g., via thermocouples), complementing widely used non-intrusive optical techniques based on soot spectral emission, which only provide data within the sooting region. A database of temperature and soot volume fraction fields is generated from detailed flame simulations that include comprehensive chemical kinetics and particle production mechanisms. This dataset is further augmented with randomized variations in spatial distribution, magnitude, and geometry, within physically expected ranges. Using this augmented dataset, a singular value decomposition (SVD) is employed to reduce dimensionality, while a QR decomposition is used to identify optimal sensor locations. This approach ensures a feasible experimental setup for obtaining complementary temperature information in soot-free regions. With these strategically placed measurements, temperature fields are reconstructed across the entire flow domain leveraging the low-rank approximation provided by the previous SVD. Subsequently, a PINN is trained on the reconstructed temperature field to solve the coupled mass, momentum, and energy transport equations, thereby inferring the corresponding velocity field. The method is demonstrated on a coflow flame from a Yale-type burner and validated on the well-known Santoro flame, where both experimental temperature and velocity data are available. Results show that SSPINNs can reliably estimate temperature from sparse measurements and accurately approximate velocity fields without the need for more sophisticated experimental techniques like particle image velocimetry. Therefore, this methodology offers a promising avenue for efficient, data-driven flow characterization in combustion systems.