How do I make the if statement not end the script when the if statement is false?

Your script isn't working because you put each subsequent if statement inside the previous one.

if script.Parent.Value.Value == 1 then
    -- Everything between this "if" and this "end" runs ONLY IF the expression "script.Parent.Value.Value == 1" is true
    if script.Parent.Value.Value == 2 then -- The code here will never run if the value is not 1 because it's inside of the first if statement
    end
    -- This is why indentation is important; it makes it obvious what block your pieces of code are in
end

This is what you want instead:

if script.Parent.Value.Value == 1 then
    print('k')
elseif script.Parent.Value.Value == 2 then
	print('l')
elseif script.Parent.Value.Value == 3 then
	print('s')
end -- Notice that there's only one "end" in this example; that's because the if statement is compound instead of nested (look up nested loops/functions for a clearer understanding of the concept)

A general example of nested code blocks:

for i = 1,10 do -- Outer loop
    for j = i,10 do -- Inner loop
	    print(i + j)
    end
    -- j is not defined out here; it is only defined INSIDE the inner loop
end