The script doesn't make the part stick to the head ?

Perhaps you could try anchoring it, welding it, then unanchoring it.

And in order for the weld to be offset from the head, you need to set it's C0 property.

local area = Instance.new("Part", player.Character.Head)
    area.Size = Vector3.new(3.45, 3.34, 0.96)
    area.BrickColor = BrickColor.new("Medium stone grey")
    area.Anchored = true;
    area.Transparency = 0
    area.CanCollide = false
    area.Shape = "Block"
    area.TopSurface = "Smooth"
    area.BottomSurface = "Smooth"
local weld = Instance.new("Weld", area)
    weld.Part0 = area
    weld.Part1 =  player.Character.Head
    wled.C0 = CFrame.new(0,0,-.8);

wait();
area.Anchored = false;

player = game.Players.LocalPlayer
Mouse = player:GetMouse()

Mouse.KeyDown:connect(function(key)
key = key:lower()
if key == "z" then
local area = Instance.new("Part", player.Character.Head)
    area.Size = Vector3.new(3.45, 3.34, 0.96)
    area.BrickColor = BrickColor.new("Medium stone grey")
    area.Anchored = true;
    area.Transparency = 0
    area.CanCollide = false
    area.Shape = "Block"
    area.TopSurface = "Smooth"
    area.BottomSurface = "Smooth"
local weld = Instance.new("Weld", area)
    weld.Part0 = area
    weld.Part1 =  player.Character.Head
    weld.C0 = CFrame.new(0,0,-.8);
    local mes = script.Mesh:Clone()
    mes.Parent = area
    local rot = script.Script:Clone()
    rot.Parent = area
    local par = script.ParticleEmitter:Clone()
    par.Parent = player.Character.UpperTorso
wait(); -- what you said to do 
area.Anchored = false;
for i = 1, 20 do
    mes.Scale = mes.Scale + Vector3.new(0.004,0.004,-0.00175)
    mes.Offset = mes.Offset + Vector3.new(0,0,-0.0375)
    wait(0.0)
end
wait(0.6)
mes.TextureId = 'rbxassetid://756808353'
for i = 1, 10 do
    mes.Scale = mes.Scale + Vector3.new(0.003,0.003,-0.002)
    mes.Offset = mes.Offset + Vector3.new(0,0,0.01)
    wait()
end
wait(0.4)
for i = 1, 10 do
    mes.Scale = mes.Scale + Vector3.new(0.02,0.02,0)
    mes.Offset = mes.Offset + Vector3.new(0,0,-0.5)
    area.Transparency = area.Transparency + 0.1
    wait()
end
par:remove()
    area:Remove()
    end
end)

look here is the full script I tried what you said but it didn't work...