We have redroot pigweed in our area, which is a wild amaranth (though other weeds have the name, too). Grass keeps it from sprouting. I've never seen it in a lawn without dead patches of grass (and rarely then, either). So, I'd go for planting it in soil without grass already in it. It grows easily (when it's warm enough; many other weeds sprout in cooler temperatures) and competes for space as well as weeds like lambsquarter, prickly lettuce and mallow. I imagine at least many of the domesticated amaranths are at least somewhat similar, from what I read about them.
If the grass hasn't sprouted yet, it's possible it may be able to grow there, as long as it sprouts first, but I'd still recommend the no-grass route.
Grass can keep a lot of things from sprouting, I've found.
You'll probably want to seed amaranth when it's warm enough to transplant watermelons (not directly after the last frost, but when it gets somewhat warmer), if the domestic kinds are like ours.