Set leaderstat "Objects" to 0 when player dies?

I see 2 errors in your script.

1st one:

if script.Parent.Parent.Parent.Parent.Character.Humanoid.Health = 0 then
    Player.Objects.Value == 0
end

You used = when CHECKING if the player's health is 0 and == when SETTING the value of Objects to 0.

= is used to SET a value and == is used to CHECK if a value is equal to another value.

2nd one:

Don't use LocalScripts to save things. Use a Script in ServerScriptStorage instead:

local DataStoreService = game:GetService('DataStoreService')
local DataStore = DataStoreService:GetDataStore('SaveSystem') -- you can name SaveSystem something else

while wait(1) do -- that way it will check every second
    game.Players.PlayerAdded:Connect(function(player)
        local leaderstats = Instance.new('Folder', player)
        leaderstats.Name = "leaderstats"
        local Objects = Instance.new("IntValue", leaderstats)
        Objects.Value = DataStore:GetAsync(player.UserId) or 0 -- it will use the value from the player's data or 0 if joined the first time
        DataStore:SetAsync(player.UserId, money.Value)
        Objects.Changed:Connect(function()
            DataStore:SetAsync(player.UserId, Objects.Value) -- that way it will save Objects whenever it is changed
        end)
    end)
    if player.Character.Humanoid.Health == 0 then
        Objects.Value = 0
    end
end

You can use Died event for your script. For example:

local Player = game:GetService("Players").LocalPlayer

Player.Character.Humanoid.Died:Connect(function()
       --your code
end)