Change Query String to pretty permalink

Try this rewrite rule in your functions.php

function string_url_rewrite() {
    global $wp_rewrite;
    add_rewrite_tag('%pet-page%', '([^&]+)');
    //In the rewrite rule you need to enter page slug and page id
    add_rewrite_rule('^Enter_Page_slug/([0-9]+)/?', 'index.php?page_id=Enter_Page_ID&pet-page=$matches[1]', 'top');

    $wp_rewrite->flush_rules(true);
}
add_action('init', 'string_url_rewrite', 10, 0);

Now you also need to change the format of URL in anchor tag. For example, your previous URL

http://www.example.com/my-account/view-pet/?pet-page=2
    

Change this URL into this

http://www.example.com/my-account/view-pet/2

To fetch value of pet-page use following method

//Storing the value of 'pet-page' in variable
$value = get_query_var( 'pet-page' );
if($value){  //checking if variable has any value or not
    echo 'This is the value '.$value;
} else {
    echo 'No Values';
}   

Note: If this don’t work then flush your permalinks. For that, go to setting -> permalink and save changes.