Did you see the section of the Parse REST API on Batch operations? Batch operations reduce the number of API calls needed to grab data so that you are not using a call for every row you retrieve. Keep in mind that there is still a limit (the default is 100, but you can set it to a maximum of 1000). That means you are still limited to pulling down 1000 rows per API call.
I can't comment on StackMob because I haven't used it. At my present job, we are using Parse and we wrote a C# app which compares the data in a Parse class with a SQL table and pulls down any changes.