Slow can reduce a Zombie's Actions
Errata has clarified that Method 2 below (rules as you would expect them to work) is correct.
Can a minion be quickened or slowed?
Yes. This can be a bit unclear because those conditions apply “at the start of your turn” and a minion can’t typically act until you use an action. Apply these conditions and any other effects that alter a minion’s number of actions when the minion gains its actions, using 2 actions and 0 reactions as the minion’s starting number. Though a minion can’t normally act when it’s not your turn, abilities that specifically grant a minion a reaction provide an exception to this (such as the Ferocious Beasts orc ancestry feat, Advanced Players Guide page 19).
There (used to be) two schools of thought on this
Method 1: "Strict RAW" The Minion gains 3 actions at the start of its turn. Minion rules state that they can only "use[sic] only 2 actions per turn and can’t use reactions". They "have" a third action they can't use. Slow 1 removed an action of their choice and they can choose to lose the un-spendable one.
Repercussions: Minions cannot use Quickened and are naturally resistant to Slow and Stun effects. Animated Zombies can use both of their Actions.
Method 2: "Rules as you might expect them to work". Minions gain a turn with two usable actions when commanded.
Repercussions: Minions can be affected by Quickened, Slow, and Stunned as normal. Animated Zombies get 1 Action even on the round they are summoned because they start their 'turn' with the Slow condition from their statblock.
Method 3: "The one we don't talk about... or, The Strictest of RAW". Minions don't have turns and never acquire Actions. They are allotted the ability to use two Actions by the Command on their commander/owner's turn, which is not theirs.
Repercussions: As Strict RAW, but Minions are completely immune to Slowed and Stunned. Also, you must also divide by zero, risking causing a portal to the Astral Plane to open us (see Bags of Holding rules) and/or may get kicked out of your game group.