I am not entirely clear about the design of the study but try this approach. First, create a data table with a data column for race/ethnic status, gender, age, and the six responses. Enter each response on a separate row. In other words, do not pre-summarize the data by "cell."
Select Analyze > Fit Model. Select the six response data columns and click Y. Select status, gender, and age. Click Macros and select Factorial to Degree. This model assumes the possibility of interactions (e.g., the effect of age might differ between sexes) but no non-linear effects. It is a good place to begin. Be sure to select the Fit Separately option in the upper right corner. Click Run.
See the Help > Books > Fitting Linear Models book. There are several related chapters dealing with these models and the options and results from the Fit Least Squares platform.
I am always amazed how Mark provides such good advice with so little input... In any case I have some additional thoughts:
1. What are the "parent-completed rating scales"?. Have you assessed this measurement system? Is there consistency within and between rater? How much variation is there within/between rater? Are you just averaging these ratings? How are you handling rater bias?
2. I'm not exactly sure what you mean by repeated measures. If indeed they are repeats (multiple data points without the treatment combinations changing), they do not add degrees of freedom. So it looks like you will have 5 DF's. As Mark points out, it would be most logical to look for linear relationships (non-linear race/ethic status is illogical) so you will have to "give up" a degree of freedom to estimate the linear effect of the covariate (no interaction).
3. I would typically recommend replicates to add DF's to investigate covariates (you will have both fixed and random effects in the model).
4. Analyze (fit model) the covariate using sequential tests (Type 1) and partial (Type 3) for the fixed effects. Turn on VIF's (add column in parameter estimates) to look for correlation among the predictor variables.