I believe that the existing answers/comments were correct at one time, but are now out of date. For both Drupal 7 and Drupal 8, this is currently possible in core. If you edit the term and expand the "Relations" section, you can select multiple parents from the multi-select input.
So for example, using the original question's taxonomy, you could edit the "Protein" term, expand the Relations section, and select both "Parent Term 1" and "Parent Term 2".
A side-note: Once you've given terms multiple parents in this way, Drupal disables the drag-and-drop mechanism for assigning hierarchy. (Since it no longer makes sense, given that the same term appears multiple times in the list.)