Using some kind of regex matching you can do this. Here's one of them. Just drop in this snippet in yout theme's
functions.php file so that it will print content of variable
$img just after first paragraph (i.e. after first occurrence of
</p> tag) in your post content.
passing current posts thumbnail / featured image value to $img will print that image after first paragraph.
// Goes into functions.php file
// Adds $img content after after first paragraph (!.e. after first `</p>` tag)
$url = wp_get_attachment_url( get_post_thumbnail_id($post->ID) );
$img = '<img src="'.$url.'" alt="" title=""/>';
$content = preg_replace('#(<p>.*?</p>)#','$1'.$img, $content, 1);
Modified this to append variable after first paragraph.