Noting that:

• Regex.run/2 returns nil if there is no match and that
• nil is falsy for the purposes of boolean evaluation and
• you can bind a variable in a match

For an optimisation I would likely put the explicit equality checks first since a cond proceeds in lexical order. Also I think you can collapse the :lat parsing into a single clause. And one last one, you can match and bind on the integer parsing too (the second last clause):

defp parseIndex(s) do
  cond do
    s == "--" ->
      {:latFrame, -1}
    s == "++" ->
      {:latFrame, 1}
    s == "-*" ->
      {:latEnd, -1}
    s == "+*" ->
      {:latEnd, 1}
    (match = Regex.run(~r/^([+-]\d+)$/, s)) -> 
      [_, n] = match
      {:lat, String.to_integer(n)}
    match?({n, ""}, Integer.parse(s)) ->
      {:gridNum, n}
    true ->
      {:gridText, s}
  end
end

What about doing binary matching directly?

defmodule Hello do
  def run(string) do
    do_run(string)
  end
  
  defp do_run("--"), do: {:latFrame, -1}
  defp do_run("++"), do: {:latFrame, 1}
  defp do_run("-*"), do: {:latEnd, -1}
  defp do_run("+*"), do: {:latEnd, 1}

  defp do_run(<<"+", number::binary>>), do: {:lon, to_integer(number)}
  defp do_run(<<"-", number::binary>>), do: {:lat, -to_integer(number)}
  defp do_run(<<"+", number::binary>>), do: {:lon, to_integer(number)}
  defp do_run(number), do: {:gridNum, to_integer(number)}
  
  defp to_integer(string), do: String.to_integer(string)
end

I like the solution with multiple functions, with a single Regex in the last one after handling all the simple cases. This is the best solution so far, thanks!

Here you have it with error handling

defmodule Hello do
  def run(string) do
    {:ok, do_run(string)}
  catch
    :throw, error -> {:error, error}
  end
  
  defp do_run("--"), do: {:latFrame, -1}
  defp do_run("++"), do: {:latFrame, 1}
  defp do_run("-*"), do: {:latEnd, -1}
  defp do_run("+*"), do: {:latEnd, 1}

  defp do_run(<<"+", number::binary>>), do: {:lon, to_integer(number)}
  defp do_run(<<"-", number::binary>>), do: {:lat, -to_integer(number)}
  defp do_run(<<"+", number::binary>>), do: {:lon, to_integer(number)}
  defp do_run(number), do: {:gridNum, to_integer(number)}
  
  defp to_integer(string) do
    case Integer.parse(string) do
      {num, ""} -> num
      _ -> throw({:not_number, string})
    end
  end
end