How can I make this nearly-finished script work?

That is totally correct; difference is that you said that it was the script's parent's child. Which means:

script.Parent.PackTitle

What you wrote:

script.Parent.Parent.PackTitle

That means it's the script's parent's parent's child.

You also did not call the function when the button is clicked.

Full script:

Table1={"262077956","262077680","262077809"}
local button = script.Parent -- change it to the text button that does this when it's clicked

function onButtonClick1()
    if script.Parent.PackTitle.Text == "Peak Performance Pack" then
        local Picked=Table1[math.random(1,#Table1)]
        script.Parent.Parent.Parent.Parent.NewItemsFrame.Card1.Image ="rbxassetid://"..Picked
    end
end

button.MouseButton1Click:connect(onButtonClick1) -- calling the function when it is clicked

Hope it helps!