Not exactly, though this was how it often worked out. The rule rather pertained – in cathedrals, collegiate churches and monasteries — to the conventual
Mass - of which there could be only one with the obligation of attending in choir. The conventual Mass was commonly a Missa Solemnis, since such places usually possessed enough ministers. Other Masses were commonly not sung simply because of time and expense, though in the wake of the liturgical movement, in a few monasteries, the custom of a daily or semi-daily private *Missa cantata * was followed.
In parishes, the distinction was not only for reasons of expense and availability of ministers, but also to mark out what would technically be THE “parochial Sunday Mass”. The distinction is harder to make with respect to the USA where the practice was for multiple hourly Masses to be offered. In some ways, all of them were “parochial”, even if they were all low Masses (though, for rubrical reasons, one had to be designated the main Sunday Mass._
This practice was in contrast to Continental Europe, where for many years, the main Sunday Mass held sway, with majority of the people attending that. Only one or two “Communion” Masses would perhaps be offered earlier.