To answer what is happening for you, when you parent the mesh object to the other armature, you change the parent. The mesh object can only be a child of one parent, so it is replaced. The heat weighting/skinning should remain intact though… however…
Rigging a character using more than 1 armature/armature modifier is actually fairly tricky and not something I would recommend unless you explicitly know why you are doing it! There is also only one scenario I’ve ever come across where it was even necessary. Most rigging can be done using one armature (sometimes using a second one as a control armature - but it doesn’t deform the mesh!). And if you’re ever using more than 2 armatures, you’re more than likely just making a mess I’m afraid
Here is a rig I recently created of Suzanne, taking advantage of dual armatures to avoid a dependency cycle. The face controls are a separate armature - they follow along with the deformations of the first armature, but can be manipulated to deform the mesh further. The key is that the two armatures don’t influence the same vertices at all. The concept behind this is actually very complex, and would require quite an indepth explanation which I will save for another time, because I am under the impression this is not what you are trying to do!
Generally, stick with 1 armature for 1 character unless you have a specific reason to use a second armature. If your character is made of multiple meshes, that’s fine, it just means you have to parent each object to the armature individually - but you’re using the same armature!
Aside from using a second armature as a control rig to directly drive the bones in the first armature (which is not common), or using ‘mesh constrained controls’ as presented in the suzanne rig I attached, I am not aware of any other valid reason to ever use more than 1 armature. If you can think of a reason however, please present it in this rigging thread because I would be very interested to hear it!
If you need further assistance, there are many rigging tutorials in my signature you can check out as well!