The data doesn’t fit that theory. Latino immigrants, for example, have historically had low levels of education prior to immigration. But for decades the trend has been for Latino incomes to converge with white incomes over a few generations. During that same time, incomes for Black people and native Americans have not converged at all.
It’s also not just a parental income thing. Whites in the bottom 20% of the income spectrum have much double the income mobility of Black people in the bottom 20%. The structural barriers to Black mobility go well beyond income: https://academic.oup.com/qje/article/135/2/711/5687353