A strict reading of the rules tells us that the hit is not a critical one.
Since the "attacker" which the description of the condition refers to may be not so clear and it is not a game term, we should use the plain English for its definition: per Oxford's dictionary an attacker is
a person who attacks somebody
Looking at the spell description, it is clear that the attacker1 is the caster, and not the spiritual weapon (emphasis mine):
When you cast the spell, you can make a melee spell attack against a creature within 5 feet of the weapon.
Per the rules of the paralyzed condition if the caster is not within 5 ft. of the target the hit is not automatically a critical hit.
This could be explained by the fact that the paralisys has a little role in damage by ranged attacks: aiming to a paralyzed target with a bow is easier, but dealing more damage is not. On the other hand, when battling in close quarters, if the target is paralyzed is easier to hit them and it is easier to deal a stronger blow, even hitting with an arrow fired by a bow in critical spots (e.g., the head).
In the case of the Spiritual Weapon spell, since the caster can control it even by the distance, the caster (hence the attacker) could be distant, hence even if it is easier to hit it is not automatic to deal more damage.
A more loose interpretation suggests that the hit is a critical one.
A DM could rule that since the target of the attack is paralyzed, it is easier both to hit and to damage them, even if the caster is not within 5 ft.
1 This Q&A deepens in the topic. See also Does making an attack with a previously cast Spiritual Weapon spell cause the Invisibility spell to end? and Can a hidden character make an attack with the Spiritual Weapon spell and still remain hidden?