I can understand how such a cute coop would be what your parents want in the yard. You live in a very warm place, Just a tarp over the top and the side that gets the most wind/rain would be enough shelter, to me. What about a portable electric net fence for ranging during the day? That way you could take advantage of the mobility of the coop you have, and your birds would not be confined all day. That would depend on having a large enough area to move them around. If you could post a photo of your Eglu showing how your birds are using it it could give us some more ideas. For a warm climate, I like a big cage instead of a closed-in house. Here is what I am using.
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This big cage probably cost less to make than your Eglu. It is chain link fence rail from Home Depot, fence corner connectors and a tarp on top and the back side. I used very light weight & cheap chicken wire due to the unavailability of better wire right now. I have surrounded it with electric net, which I can move to another area or store in the winter If I want.