The `{brandt}`

package author states

I added it to to the function output to remind persons what hypothesis they are testing because it is often not clear what the alternative (H_A and the null hypothesis (H_0) is. So it just tells you what the null hypothesis is and nothing about the acutal result. p < 0.05 means that H_0 can be rejected.

So in [S/O OP example] the parallel regression assumption does not hold. In generell[sic]: p-value of omnibus >= 0.05 => holds, p-value < 0.05 => does not hold (assumption: \alpha-value of 0.05).

Benjamin Schlegel (User Benjamin Schlegel - Cross Validated), Checking parallel regression assumption in ordinal logistic regression, URL (version: 2020-10-22): r - Checking parallel regression assumption in ordinal logistic regression - Cross Validated

Keeping the nulls straight is hard. (I got it backwards in my answer, now edited). Also confusing is the use of p-value and probability interchangably. In McNulty's example, everything is > 0.05, meaning reject the null and conclude that the parallel regression assumption does not hold if Schlegel's intepretation of the function he wrote is correct.

The original Brant paper is JStorWalled, so I can check it. `{car}`

has an implementation, and checking against that might clarify things.