I'm an employer. He is within his rights to use parental leave, obviously, but ensure you are au-fait with the rules. Parental leave is limited to a maximum of 18 weeks per child up to their 18th birthday. You should, therefore, find out from their previous employer if they have previously used any parental leave there, to keep a record of it.
Then, it must be taken in whole weeks, not individual days, unless your employer agrees otherwise. At this point, check your company's employee handbook and the contracts to see whether there are any specific references to this.
Our handbook says "If you are entitled to take parental leave... you should discuss your needs with a Director, who will identify your entitlements and look at the proposed leave periods dependent upon your child's particular circumstances and the operational aspects of the business".
The rules say that leave can't be postponed/delayed if the employer doesn't have a "significant reason", e.g. it would cause serious disruption to the business. Therefore you can postpone it if it does cause issues. If you do postpone it as employer you have to write to them within 7 days of the original request explaining why, suggest a new start date (within 6 months of the requested date) and you can't amend the amount of leave being requested.
So, ultimately, if agreeing to it creates a massive problem, you can tell him no and suggest an alternative date but firstly, he must properly request it (not demand or expect it) and you must respond within the rules.
Sorry if this has all been said elsewhere - I started typing it ages ago and got waylaid!