I'm trying to get the property of an item inside part inside a model, Please help?

Try This:

Lighting = game:GetService('Lighting') -- This line isn't needed though.
ReplicatedStorage = game.ReplicatedStorage
parts = game.Workspace.ArenaLights:GetChildren() 
--------------------------------
game.ReplicatedStorage.LightingEvents.ArenaLightsEvent.OnServerEvent:Connect(function(player,arenalight)
    if arenalight == "BrightWhite" then --White Lighting
        for i,v in pairs(parts) do
            v.PointLight.Color = Color3.new(255/255,255/255,255/255)
            wait(0.01)
        end
        wait(0.01)
    elseif arenalight == "White" then
        for i,v in pairs(parts) do
            v.PointLight.Color = Color3.new(255/255,255/255,255/255)
            wait(0.01)
        end
        wait(0.01)
    elseif arenalight == "Blackout" then
        for i,v in pairs(parts) do
            v.PointLight.Color = Color3.new(0/255,0/255,0/255)
            wait(0.01)
        end
        wait(0.01)
    end
end)

If I understood correctly you want to change all the PointLights inside of all the Parts inside of ArenaLights.

The simplest loop is just using a i, v pairs loop. This will loop through a list of items, in this case a list of all the children in the ArenaLights object, and give you the index (number) of the child and the child itself.

for index, child in pairs(game.Workspace.ArenaLights:GetChildren()) do
    -- Code in here will run for every single child independently.
    -- Example:
    child.PointLight.Color = Color3.fromRGB(255, 255, 255)
end

The correct usage is using i,v instead of index, child. I fixed your code for you, but it'd be good practice to try to fix it yourself using the above information. I've also used Color3.fromRGB() instead of Color3.new() because it's easier to work with.

Lighting = game.Lighting
ReplicatedStorage = game.ReplicatedStorage

--------------------------------

game.ReplicatedStorage.LightingEvents.ArenaLightsEvent.OnServerEvent:Connect(function(player,arenalight)
    local children = game.Workspace.ArenaLights:GetChildren()
    if arenalight == "BrightWhite" then --White Lighting
        for i, v in pairs(children) do
            v.PointLight.Color = Color3.fromRGB(255, 255, 255)
        end
        wait(0.01)
    elseif arenalight == "White" then
        for i, v in pairs(children) do
            v.PointLight.Color = Color3.fromRGB(255, 255, 255)
        end
        wait(0.01)
    elseif arenalight == "Blackout" then
        for i, v in pairs(children) do
            v.PointLight.Color = Color3.fromRGB(0, 0, 0)
        end
        wait(0.01)
    end
end)