You're definitely right in the 120gal part! A 4'2'2' is the minimum for any adult bearded dragon.
Though I'm a bit confused on your "they need wood or mesh enclosures"? Bearded dragons can definitely be kept in tanks (as long as there is ventilation ofc, as with any animal), though wood cages can work as well. And I'm not sure what a "mesh enclosure" is? Really the cage can be made out of any non-toxic thing secure enough so they don't escape, that has ventilation, and holds in heat & humidity to the right amount.
To the OP, I think people gave some great suggestions above so I don't have much to add. But if I may ask, why don't your parents like the idea of lizards? What specifically about them puts your parents off >.>
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they have really bad eyesight and tend to rub there noses ageinst the glass. 'Mesh' as in, a screen diy kind of cage.























