As you use an arrow function, your this doesn't refer to the object.
Use the event.target property
    <div class="control-group">
      <label for="name" class="control-label">
              <p class="text-info">Alex<i class="icon-star"></i> </p>
            </label>
      <input type="text" class="edit-input" />
      <i class="fa fa-pencil edit" aria-hidden="true">edit</i>
    </div>
    <script>
      getSelection = (e) => {
        //console.log("this", this); //valid but empty json
        console.log("e.target.parentNode", e.target.parentNode); //<=undefined
      }
      document.querySelector('i.edit').addEventListener('click', getSelection, false);
    </script>