Track custom conversion with Acuity Scheduling

Site URL: https://dodenhof.de/posthausen/service/terminvereinbarung/brautmoden-beratung/

We use the Acuity Scheduling iframe e.g. at https://dodenhof.de/posthausen/service/terminvereinbarung/brautmoden-beratung/. We would like to track conversions and found https://support.squarespace.com/hc/en-us/articles/4410642724365#toc-technical-details. To have some reliable conversion tracking, we would like to publish a conversion message to the outer parent frame, so that it can be picked up from there and e.g. forwarded to Google Tag Manager. By doing that, we would act in first party context and tracking would be more reliable compared to cross domain solutions..

I embedded some code in the "Custom Conversion Tracking" integration, but it seems there is not chance to communicate with the outer parent frame (our original website), or am I doing something wrong?