In Teamgames the idea is that you dont make a composition (meaning: different types of units). You only focus on one unit (Paladin or War Elepehants) and do what your civ does the best, and your partner makes something that complements that. So the perfect 2v2 idea is always to pick a cav civ and an archer civ, because Paladins + Arbalesters basically beat everything. The combo of a strong cavalry unit plus a strong archer unit complements each other perfectly. In some cases you could think about cavalry + cav archers, if your dad really wants to stay on Mongols.
With that in mind, your best composition with Persians + Mongols is when you go Paladins or War Elephants (and only that!) and research all Blacksmith upgrades for that and Bloodlines and Husbandry (and Mahouts in the case of War Elephants of course), and your dad goes only Mangudai with all upgrades. Then your dad adds some siege (like Siege Rams) and you both can bring some Trebuchets. And this all together is the best composition you could do as Persians + Mongols. But the main idea is that you dont waste resources and population space to skirmishers and such, and only do either knights or elephants, and your dad accompanies that with Mangudai, and then you fight as a team.