12V fan does not work when in parallel but does in series
The TEC has a very low resistance, comparable to or lower than the internal resistance of the solar panel. This means that when it is connected across the solar panel, the voltage across it drops too low to run the fan. On the other hand, when you connect it in series with the relatively high-resistance fan there is plenty of voltage across the fan to run it since there is very little voltage drop across the TEC. Unfortunately this means that more power is being used by the fan than the TEC, so not much cooling will happen.