Cacti and succulents of this type prefer to stay fairly dry between waterings, since they store so much moisture in their stems/leaves. The soil should be allowed to dry several inches beneath the surface before being drenched well enough to drain out of the pot's bottom holes. How often watering will be needed is hard to predict as it depends on several factors like light level, ambient temperature and humidity, airflow, if the plant is in active growth, and how old (and therefore compressed) the potting mix is. While saucers are usually needed to catch excess moisture draining after a thorough watering, they should be emptied right away so the moisture doesn't re-absorb into the soil, causing root rot. The alternative is to water plants over the sink and let them drip-dry there for a few minutes before returning them to the saucer. You may be doing this already, but diagnosing the cause of plant collapse can be difficult, since symptoms are sometimes delayed. When cacti die back, over-watering is the most likely suspect.

The sunny window exposure and 78-degree temperature should not be a problem.

The cactus pad that is discolored at its base and fallen is rotted and needs to be removed. That said, its small side growths look fairly intact, as does the green pad that has already fallen off. Cutting off the healthy portions, letting the wounds air-dry as-is for a few days so they form a callous, and setting the pieces on top of (or barely buried into) fresh cactus potting mix may encourage them to root and survive. For the short term, you do not need to worry that they currently have no roots. If they do well, rooting and forming new pad growth may take weeks. Shriveled sections of pad, though, probably won't make it, and any browned or blackened areas should be thrown out. The other flopped pad that's entirely gray-brown is dead and should be removed.

If the soil in the cactus pot is several years old, it may be retaining moisture for too long, especially if the frequency of watering you're doing happens to be greater than it was receiving when your folks were home. If that's the case, let it get drier between waterings, and suggest they repot the plant into fresh potting mix when they're back. Components in potting mixes used to add airiness naturally degrade and compress over time, so after awhile, it's best to refresh the mix to maintain good oxygenation for the roots.

We can't quite tell if the pale material on the cactus pad's base is natural corky growth (which develops from the bottom up on some species as the tissues age) or possibly a very heavy scale infestation. Scale are sucking insects which don't move (except for their just-hatched juvenile stage, which is very tiny) and essentially glue themselves to the plant's surface. You can usually carefully flake them off with a fingernail or toothpick, though, so see if this substance can be removed. If it is normal tissue growth, it won't come off, at least not without exposing the juicier tissues underneath and causing a wound. If it is scale, this population looks so heavy that it's probably been building gradually for many months or years, and the plant is much more easily replaced than treated at this point. In that event, though, there's no harm in letting it go untreated a little longer until your folks get home and can decide what they want to do about it.

The mushrooms along the edge of the pot do suggest excessive moisture is a problem, though their presence likely isn't a direct concern as it probably won't be infecting the cactus. Many soil-dwelling fungi only feed on organic matter that is already dead (such as roots that died from wet conditions, or the components of the potting mix itself). It wouldn't hurt to remove them, but their spores disperse pretty quickly so the mushrooms may already be on their way to collapsing and disintegrating on their own.

Miri