We study multilingual AMR parsing from the perspective of knowledge distillation, where the aim is to learn and improve a multilingual AMR parser by using an existing English parser as its teacher. We constrain our exploration in a strict multilingual setting: there is but one model to parse all different languages including English. We identify that noisy input and precise output are the key to successful distillation. Together with extensive pre-training, we obtain an AMR parser whose performances surpass all previously published results on four different foreign languages, including German, Spanish, Italian, and Chinese, by large margins (up to 18.8 Smatch points on Chinese and on average 11.3 Smatch points). Our parser also achieves comparable performance on English to the latest state-of-the-art English-only parser.