Putting your money in the same account as a parent could cause many problems, with very few benefits.
One of you would have to claim the dividends and capital gains that the fund might earn during the year. That person might have to pay taxes on those earnings. You would have to find some way of figuring out how to split the costs, and somebody would have to reimburse the other.
If one person wanted to sell, figuring out which shares to sell would be much more complex.
If these are retirement accounts, which have maximum limits based on income, and the use of other retirement accounts, there is no way to co-mingle the funds.
Even if it was possible to combine the funds, the reality is that two people decades apart have different investment goals, and risk tolerances, so the types of investments that a great for one, are very poor for the other.
The only benefit is that an existing account would already have more than the minimum investment, so some investments would be easier to make. Also some investments have lower fees if you meet specific investment thresholds.
If the fund increases in value by 10% in a year, it doesn't make a difference if the value at the start of the year was 10K or 100K. The rate is the same.
The benefits are minor and few; the drawbacks are many; and some situations make it impossible to co-mingle the funds.