This paper investigates a novel downlink symbiotic radio framework enabled by the pinching antenna system (PASS), designed to enhance both primary and secondary transmissions through reconfigurable antenna positioning. This reconfigurability introduces additional degrees of freedom for adaptive pinching beamforming, thereby enabling constructive signal enhancement and interference suppression tailored to the locations of the backscatter device, the Internet of Things (IoT) receiver, and the primary receivers. To fully exploit these benefits, we formulate a joint transmit and pinching beamforming optimization problem that maximizes the achievable sum rate while satisfying the IoT receiver's detection error probability constraint and feasible deployment constraints for the pinching antennas. The resulting problem is inherently nonconvex and highly coupled. To address this challenge, we develop two complementary solution approaches. The first is a learning-aided gradient descent method, where the constrained optimization is reformulated into a differentiable form and solved through end-to-end learning. In this approach, the pinching antenna position matrix is reparameterized to automatically satisfy minimum spacing constraints, while transmit power and waveguide length limits are enforced via projection and normalization. The second approach is an optimization-based successive convex approximation-particle swarm optimization method, which first determines the transmit beamforming solution using successive convex approximation and subsequently optimizes pinching beamforming via a particle swarm optimization search over candidate pinching antenna placements.