While I have not done what you are trying to do. what you have done the first part of generating a genotype for each individual.
The next step I would suspect would be to phase these genotypes into the two haplotypes in each gene (for both children and parents). I think you will not be able to do this in Galaxy (but I may be wrong). Programs like shapeit or Beagle would do the job easily, they can use massive human haplotype reference panels to help do this with a high degree of accuracy (both programs output confidence metrics).
You would want to do this for each gene separately but including all individuals ( I know shapeit will take .vcf files). Ones you have the haplotypes of all the individuals involved I guess that it is trivial to identify the parents.
PS If you do find a way to phase genotypes within Galaxy please tell me as this is something I might be interested in doing in the future.