Your issue is your envy. Now, your envy is not your companies problem; it is your family's problem.
So an answer to this has to be broken into two parts. The family problem, and the workspace problem.
First, I'll pretend there is no family involved, which can illuminate the situation.
You have a job where you are underpaid. As part of this job, you are also expected to be able to "buy in" to the business at some future date. This "buy in" option may be at lower than market rates, so may have some value.
In addition, as a future executive, how the company works right now is something of a concern. It has another employee whose performance is not matching their salary who also has "buy in" options.
Another problem is that your company has a policy to pay both positions the same money.
For your salary, the first question you should ask is if you have leverage. Is your compensation low enough that you can walk and get a better deal? Your pay is low; does the opportunity to buy-in later make up for it?
The second question is, do they have leverage? You can simply state you are no longer able to travel because of family demands. If you are that great a deal, will they be able to coerce you? Similarly, you can ramp down any overtime and only commit to more reasonable deadlines with lots of padding in case things don't go as planned, reducing your stress.
If they state they need you to get things done as fast and travel just as much, but cannot increase your base salary, state under what terms you would be willing to continue to travel and what terms you are wiling to do overtime. Then when giving estimates for tasks, give the slow and steady rate and how much time you can cut off that with how much overtime.
Without changing your base salary, this can permit higher compensation. This can be used to get around the policy that your two salaries are the same.
The separate problem is that your coworker isn't earning their keep. You can manage upwards and find a way to move more tasks over to this coworker that are harder and more solid in result. Rigorous QA is an important, hard and tedious task that someone with lots of free time could study up on, for example. You have resources being wasted on someone under-performing that you cannot fire; there are ways to solve this that don't involve firing the person in question.
If you are going to be an executive for your business, learning how to get more out of under-performing workers is important. Simply firing them is easier, but replacing them is a pain that can be avoided if you can just up their productivity. View the underperforming colleague as an opportunity to up your management chops.
Finally, note that your coworker seems to be in a supervisory role. One thing that managers value highly is supervision; they do that, and they like valuing what they do highly, and people supervising for them makes their lives easier. Your execution role may not be valued as highly as it is possible the head of the company doesn't know what it involves.
Alternatively, you may have the same problem; viewing your task as harder because you are familiar with it, while discounting the difficulties of supervision.
Family wise, you have a problem where the patriarch views the employment of the kids as patronage not as a job. They seem to be expected to contribute what they are capable of, and get back an even share.
Solving this problem is an interpersonal skills issue, and will depend on the dynamics of your family and piles of history that isn't appropriate to relay here.
You can try to get the patriarch to treat their position of equal salary as a fixed thing, and compromise on bonuses/overtime/travel/etc. Or some other wya to make the higher compensation of your role be "not salary". This avoids colliding with your patriarch head-on. You can justify it using market rates or anything else, but providing an "exit" so that the fixed position the patriarch took doesn't technically change you may be able to fix your issue without insulting your father's pride or sense of fairness.