That’s the main point. You cannot add sub-domain to your Cloudflare account.
Firstly, you have to add the naked domain like domain.com.
Therefore, you create an A record for sub.domain.com or sub.sub.domain.com.
Nevertheless, for that “deep” sub-domains with Xth and so on level, I am afraid you would also have some issues in terms of the SSL certificate as far as below article explains more in detail:
Meaning, they are managing their DNS records by their Cloudflare account to which, you as obviously, again, do not have the access, right?
Could you ask them to add your e-mail / account under their main Cloudflare account, so you could, at least, manage the needed?
See below article for more information about Multiple-user accounts on Cloudflare:
Otherwise, you could ask them to add a new DNS record for that sub.domain.com sub.sub.domain.com, therefore depending which level that sub-domain is, I am afraid they would have to purchase an SSL certificate using Advanced Certificate Manager from below link in case of 
cloud (Proxy mode for that sub or sub.sub domain):
Nevertheless, if you have an SSL certificate on your origin host / server which covers that sub-domain or sub.sub-domain, they could set it to 
(DNS-only) - but in that case, you lose all the benefits from Cloudflare and you actually do not need Cloudflare in that terms as well (from my understanding and my point of a view).
Could be I am wrong about something.
Kindly and patiently, wait for another reply if so.