This is unlikely to be true. Antibodies raised by MMR vaccine (and BCG etc) are not cross reactive against Sars-cov-2.
Anecdotal evidence showed a negative correlation between Bacille Calmette-Guérin (BCG) vaccination and incidence of COVID-19. Incidence of the disease in children is much lower than in adults. It is hypothesized that BCG and other childhood vaccinations may provide some protection against...
journals.plos.org
The study people are referring to (link below) finds, maybe, a correlation but that's a lot different from showing MMR vaccination is causally related to reduced severity of COVID. The groups they are comparing are small and not matched for important factors (eg age) that significantly affect COVID disease course. On top of that the authors haven't looked at patients who got MMR as adults to see the effect on COVID, which would be important to know.
The flaws in this study, combined with the lack of translational evidence for a biological effect, and flaws in the previous work looking for protective effect of BCG on COVID, make it unlikely that MMR vaccination is a useful prophylaxis for COVID - especially when actual biologically active vaccines are so close.
(From what I can tell the lead author runs an animal shelter and the other authors are generally not experts in this subject)
The measles-mumps-rubella (MMR) vaccine has been theorized to provide protection against coronavirus disease 2019 (COVID-19). Our aim was to determine whether any MMR IgG titers are inversely correlated with severity in recovered COVID-19 patients previously vaccinated with MMR II. We divided 80...
mbio.asm.org
There is some evidence that childhood vaccinations offer benefits in immunity to other diseases but that is pretty fuzzy and not a rationale for using them in COVID.