In my case, the following solved this problem
HttpEntity<MultiValueMap<String, Object>> requestEntity = new HttpEntity<MultiValueMap<String, Object>>(parts, requestHeaders);
RestTemplate restTemplate = getRestTemplate();
restTemplate.getMessageConverters().add(new FormHttpMessageConverter());
restTemplate.getMessageConverters().add(new MappingJackson2HttpMessageConverter());
return restTemplate.postForObject(URI, requestEntity, APIResponseUpload.class);
However, according to the docs for Spring, this it shouldn't have been necessary to manually set the message converters.