Recent progress in task-oriented neural dialogue systems is largely focused on a handful of languages, as annotation of training data is tedious and expensive. Machine translation has been used to make systems multilingual, but this can introduce a pipeline of errors. Another promising solution is using cross-lingual transfer learning through pretrained multilingual models. Existing methods train multilingual models with additional code-mixed task data or refine the cross-lingual representations through parallel ontologies. In this work, we enhance the transfer learning process by intermediate fine-tuning of pretrained multilingual models, where the multilingual models are fine-tuned with different but related data and/or tasks. Specifically, we use parallel and conversational movie subtitles datasets to design cross-lingual intermediate tasks suitable for downstream dialogue tasks. We use only 200K lines of parallel data for intermediate fine-tuning which is already available for 1782 language pairs. We test our approach on the cross-lingual dialogue state tracking task for the parallel MultiWoZ (English -> Chinese, Chinese -> English) and Multilingual WoZ (English -> German, English -> Italian) datasets. We achieve impressive improvements (> 20% on joint goal accuracy) on the parallel MultiWoZ dataset and the Multilingual WoZ dataset over the vanilla baseline with only 10% of the target language task data and zero-shot setup respectively.