Mplus does not include residual variances for Poisson regression with a dependent variable that is a count variable - this is in line with regular Poisson regression analysis where such a residual is absent. The Theta parameterization is not in effect for count outcomes. You can, however, in Mplus define a residual by defining a factor that influences the dependent count variable, e.g. when regressing y on a covariate x,
I am estimating a parallel process latent growth model with four outcomes, over three timepoints. Three outcomes are continuous and the fourth is a count of conduct problems (from 0 to 7). This is my input:
You could try using only 1 factor for each of the 3 time points and let the different loadings capture the different sizes of the residual correlations among the 3 processes. So 3 dimensions instead of 9. Make sure the 3 factors are uncorrelated with each other.