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If you would like a component to be displayed on only one page, that can be done by adding the following code to your theme's template.php file:
<?php if (return_page_slug()=='PAGE') get_component('COMPNAME'); ?>
Don't forget to replace PAGE with your page's slug and COMPNAME with the component's name.
Now If you want a component to be displayed on pages of a specific parent page, you can do it in a similar way:
<?php if (get_parent(0)=='PAGE') get_component('COMPNAME'); ?>
Again, don't forget to replace PAGE with your page's slug and COMPNAME with the component's name.
If you would like to exclude a component from a page you could do that by replacing “==” with “!=” in the code above.