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In the autonomous driving environment, each vehicle performs numerous signing and verification while sending and receiving BSMs (Basic Security Messages) in real-time. We present an optimized CRYSTALS-Dilithium software which can quickly process each signing and verification in parallel using embedded Graphic Processing Unit. For efficiency, we propose several optimization techniques such as dummy operation-based warp divergence reducing technique, parallel implementation NTT (Number Theoretic Transform)-based polynomial multiplication, optimization of rejection sampling process using rejection sequence table, and so on. The proposed CRYSTALS-Dilithium software provides a performance improvement of up to 19.41 times compared to the Dilithium Software on CPUs on NVIDIA Jetson AGX Xavier which is an off-the-shelf autonomous vehicle OBU (On-Board Unit).