The thing is that you are using css classes for targeting your datetime pickers, and you are giving both the start and end dates the same class name: timepicker1.
Try changing the references to the ID instead of the class data-target="#timepicker1", for which you'll have to add the id="timepicker1" to the start_time div:
<div class="input-group date timepicker1" data-target-input="nearest" id="timepicker1">
<input type="text" name="start_time" class="form-control datetimepicker-input" data-target="#timepicker1"/>
<div class="input-group-append" data-target="#timepicker1" data-toggle="datetimepicker">
<div class="input-group-text"><i class="far fa-clock"></i></div>
</div>
</div>
then change the references at the end_time to use ID too data-target="#timepicker2":
<div class="input-group date timepicker1" id="timepicker2" data-target-input="nearest">
<input type="text" name="end_time" class="form-control datetimepicker-input" data-target="#timepicker2"/>
<div class="input-group-append" data-target="#timepicker2" data-toggle="datetimepicker">
<div class="input-group-text"><i class="far fa-clock"></i></div>
</div>
</div>
I guess you won't need to change the JS code, but if it still doesn't work, you could try initializing both pickers by their id too:
$('#timepicker1, #timepicker2').datetimepicker({
multitime: true,
use24hours: true,
format: 'HH:mm'
});
Hope this works!