The edge artificial intelligence (AI) applications in next-generation mobile networks demand efficient AI-model downloading techniques to support real-time, on-device inference. However, transmitting high-dimensional AI models over wireless channels remains challenging due to limited communication resources. To address this issue, we propose a parametric-sensitivity-aware retransmission (PASAR) framework that manages radio-resource usage of different parameter packets according to their importance on model inference accuracy, known as parametric sensitivity. Empirical analysis reveals a highly right-skewed sensitivity distribution, indicating that only a small fraction of parameters significantly affect model performance. Leveraging this insight, we design a novel online retransmission protocol, i.e., the PASAR protocol, that adaptively terminates packet transmission based on real-time bit error rate (BER) measurements and the associated parametric sensitivity. The protocol employs an adaptive, round-wise stopping criterion, enabling heterogeneous, packet-level retransmissions that preserve overall model functionality but reduce overall latency. Extensive experiments across diverse deep neural network architectures and real-world datasets demonstrate that PASAR substantially outperforms classical hybrid automatic repeat request (HARQ) schemes in terms of communication efficiency and latency.