If we delete all the rows in a table by mistake (!) on Parse, is there a way to recover from it at all? Are there automatic backup's that we can restore from?
Parse itself does not have tools for recovery of deleted items. Once you delete something, it is gone.
However, if you click Settings, there is an option on the left called "Export Data", which will email you a zipped JSON file with all of your data. I recommend that you do this regularly. Then, if you somehow accidentally delete all of your data, you can at the least restore to the latest export by clicking the "Import Data" button at the bottom of the data browser and selecting the downloaded JSON file.
Keep in mind that the links Parse sends you with your data are temporary, so be sure to download and store the file.