The answer to "Can I leave me car here?" is shown as "Kann ich mein Auto hier lassen?" But should this not be "Kann ich hier mein Auto lassen?" as time, manner and place are supposed to come before the direct object noun.
For the next phrase "Can I leave me car on the parking lot?" the answer is shown as "Kann ich auf dem Parkplatz mein Auto lassen?" which does put the direct object noun after the place.
You're perfectly right CMcCullough, here the first question could also be with "hier" before "mein Auto", both ways are fine. As many times in German, several orders are possible, and the difference is subtle and more a matter of emphasis or of viewpoint depending on the situation. In any case we'll make sure all options are available for this phrase for our next update, thanks for commenting!