The conditions are not linked so the Barbarian would be paralyzed but not frightened
This answer is incorrect as the lore found in Illustro's answer shows that they are linked. The following is an examination of the wording without reference to the lore.
The Dreadful Glare feature simply states:
[...] If the target fails the saving throw by 5 or more, it is also paralyzed for the same duration [...]
There is no connection made between the frightened condition and the paralyzed condition. Additionally I believe that "for the same duration" means "until the end of the Mummy's next turn" not "for the duration it is frightened", but I can certainly see that interpreted either way.
Compare this to the fear spell which states:
[...] While frightened by this spell, a creature must take the Dash action and move away from you by the safest available route on each of its turns [...]
Here the requirement to take the Dash action only exists while frightened. There is no such linking in the Mummy's ability so in this case the Barbarian would be paralyzed for failing the save by 5+ but would not be frightened due to their Mindless Rage feature.
Another example of linked conditions is the flesh to stone spell which states:
[...] A creature restrained by this spell must make another Constitution saving throw at the end of each of its turns [...]
Here the Constitution saves (and their effects) are only made if the creature is restrained by the spell itself. The Dreadful Glare feature uses neither of these more "obvious" linking methods which leads me to believe that they are not linked.
Finally, the Dreadful Glare feature is similar to the Ghost's Horrifying Visage feature:
Each non-undead creature within 60 feet of the ghost that can see it must succeed on a DC 13 Wisdom saving throw or be frightened for 1 minute. If the save fails by 5 or more, the target also ages 1d4 × 10 years.
The question "How does a ghost's Horrifying Visage aging interact with immunity to the frightened condition?" currently has a decently upvoted answer by BlueMoon93 arguing that the frightened condition and the aging are not linked. As Dreadful Glare uses similar wording, I would assume it is likewise, not linked.