Dear Johanna

Yes, that is essentially how group comparisons work within SPM. More generally, this is known as the summary statistic approach to mixed effects analyses of group data; in the summary statistic approach one computes mixtures or contrasts of parameter estimates at the first (within subject) level. These contrasts are used as summaries of a treatment effect of interest and passed to the second (between subject) level to make an inference, in relation to intersubject variability.

You are absolutely right that this inference means that the difference in group means is compared (in relation to the variability about that mean from subject to subject). The group mean difference is accessed through a contrast testing for the difference in means and is the contrast image produced at the second level.  Note that your parametric maps are really contrast images; in other words, linear mixtures or contrasts of parameter estimate maps.

I hope this helps.

With very best wishes,

Karl