Skimmed a bit over some of the responses, but having been in a similar position no OP I don't think your DP is being unreasonable.
Legally - pp are correct your DP has no legal right to know.
Morally - it's a bit weird, especially as they had to know all about you (same as with my now DH - every last detail must be known).
The gut feeling of is something being hidden. Nought about control, just that gut feeling of something off, may not be nothing, but when it comes to your DCs it's hard to ignore.
In the end of many weird situations (not going into them as potentially outing) - nothing major but just set my I'm not sure of this...... gut something not quite right ... I employed a Private Investigator.
Found the ex was taking my DS to a house with people who had multiple identities (seriously loads of names) and low level criminal convictions - and often leaving him there without him, with the on off GF - who also had let's call it an interesting record. Doesn't sound like much, but it explained a lot and explained why I couldn't have a name, or wasn't "allowed" to know where my DS was at the weekend.
It was enough to change the custody arrangements back in court.
See how it plays out, if there is something making your DP uncomfortable or if things with the DC don't seem right - there are ways to check, that at the very least will set minds at rest or at worst evoke action to protect.