Since you're running PHP, it's as simple as wrapping all of that within a PHP if-block:
<?php if( get_field('field_link') && get_field('field_image') ): ?>
<a href="<?php the_field('field_link'); ?>">
<img src="<?php the_field('field_image'); ?>"></a>
<?php endif; ?>
Also, if you're unfamiliar, the above uses PHP's Alternative Syntax for control structures.
Edit: I understand now this is using the actual plugin Advanced Custom Fields and updated my code.