JSFiddle - React, Tailwind, and code Playground

HTML

<link rel="stylesheet" href="http://code.jquery.com/ui/1.9.1/themes/base/jquery-ui.css" />
<script src="http://code.jquery.com/ui/1.9.1/jquery-ui.js"></script>
<input type="text" id="datepicker1" name='date1'/>
<input type="text" id="datepicker2" name='date2'/>

CSS

.just-yaer .ui-datepicker-prev,
.just-yaer .ui-datepicker-next,
.just-yaer .ui-datepicker-month,
.just-yaer .ui-datepicker-calendar {
    display: none;
}
.just-month .ui-datepicker-prev,
.just-month .ui-datepicker-next,
.just-month .ui-datepicker-year,
.just-month .ui-datepicker-calendar {
    display: none;
}

JavaScript

$(function() {
    $('#datepicker1').datepicker( {
        changeMonth: false,
        changeYear: true,
        showButtonPanel: true,
        dateFormat: 'yy',
        showAnim: '',
        beforeShow: function( input, inst){
          $(inst.dpDiv).addClass('just-yaer');
        },
        onClose: function(dateText, inst) { 
            var year = $("#ui-datepicker-div .ui-datepicker-year :selected").val();
            $(this).datepicker('setDate', new Date(year, 1));
            $(inst.dpDiv).removeClass('just-yaer');
        }
    });
});

 $(function() {
    $('#datepicker2').datepicker( {
        changeMonth: true,
        changeYear: false,
        showButtonPanel: true,
        dateFormat: 'MM',
        showAnim: '',
        beforeShow: function( input, inst){
          $(inst.dpDiv).addClass('just-month');
        },
        onClose: function(dateText, inst) { 
          	var month = $("#ui-datepicker-div .ui-datepicker-month :selected").val();
            $(this).datepicker('setDate', new Date(month, 1));
            $(inst.dpDiv).removeClass('just-month');
        }
    });
});