PHP mail sending empty mails

// Your required fields:
$fields = ['parent', 'child', 'contacts', 'email', 'bbd', 'city', 'mess'];
// Your data is collected here:
$data = [];
// Any errors are collected here:
$errors = [];

// Loop to check your required fields:

foreach($fields as $field) {
    // If value exists, add to $data:
    if(!empty($_POST[$field])) {
        $data[$field] = $_POST[$field];
    }
    // Else add error:
    else {
        $errors[] = 'Missing field: ' . $field;
    }
}

if(empty($errors)) {
    // No errors, send your email
    // You can use "Vanema nimi {$data['parent']}...",
    // ... otherwise: extract($data) to use $parent etc.
}
else {
    // You could report those errors, or redirect back to the form, or whatever.
}