Minimum hours required to be paid regardless of hours worked my employer?

I'm a nanny and I'm currently working part-time with an average anywhere from though and a 15 hours to the higher-end 30 depending on the parents schedule. But there's going to be some unusual situations coming up with mom having a new baby towards the end of the year and I noticed that one of the parents hours is being cut back as far as what the required to work for the day. They have a schedule of 5 p.m. to 1 a.m. but yeah their parent gets home anywhere between about 5:30 or so. I have a quite a bit of distance to drive on a normal day out traffic which is my decision but I'm worried that they could do what they've done in the past and asked me to come in for just like an hour and a half or two hours. And when it takes me an hour I'm more to get home or get there in the morning with rush hour traffic it's just kind of silly to only get paid for let's say an hour and a half or two hours. although I don't want to run the risk of losing my job by saying repeatedly that I can't do those hours but isn't it required by the employer to give you some sort of minimum amount per day? I know when I worked in management we are required to pay a minimum of three to four hours or something a day regardless of how many hours they actually worked so I remember my supervisor saying well go ahead and keep them there and give them whatever jobs you can because if we're going to have to pay them by law might as well have them there physically. I know this is not a corporate job but I still think fairness should be taking into account here.