The correlation in Europe tend to be between parental education level and child education level. The correlation between parental income and education level of the child is weaker.
This is true in the United States (there is a very strong correlation between wealth and education level, but the correlation between parent's education level and child's education level is stronger than the correlation between parent's wealth and child's education level; IIRC, there still some evidence that the wealth plays some role independent of parent's education level in determining child's education level, but its a smaller role.)