You can include a main effect of GED and an interaction between GED and parental income without including parental income. This is equivalent to encoding the restriction that for those without a GED, there is no relationship between parental income (PI) and income, which is a strong assumption to make. Here's the model you would fit:
E[income|GED, PI] = \beta_0 + \beta_1 GED + \beta_2 GED \times PI
You would use the mean-centered version of PI. The interpretation of the coefficients would be the following:
- $\beta_0$ - the expected income for those without a GED
- $\beta_1$ - the difference between the expected income for those with a GED and those without a GED for those with an average level of PI
- $\beta_2$ - the slope of PI on income for those with a GED
Typically, a model would also have a main effect term for $PI$, which represents the effect of $PI$ on income for those without a GED, and which changes the interpretation of $\beta_2$ to be the difference between the effect of $PI$ on income for those with a GED and the effect of $PI$ on income for those without a GED.
In R, you would fit this model using the following syntax:
fit <- lm(income ~ GED + GED:PI)
You must use a robust standard error for the tests to be valid because you are by definition explaining more variance in income for those with a GED than for those without, so the residual variance differs between groups. To estimate the robust standard errors, you could use the following syntax:
lmtest::coeftest(fit, vcov. = sandwich::vcovHC)
I don't recommend you fit a model like this unless you were unable to measure PI for those without a GED. The assumption you make is a strong one and almost certainly false, so the rest of the model may not be valid.