After the mother dies, while the father is still alive, why is Amavasya tarpanam not performed for the mother alone? The mother is also a ‘pithru’ (ancestor), please explain.

Tarpana is done as a substitute for ‘Shraddha’. In that, Shradha, ‘homam’ (fire ritual) is done only for the father. There is no homam for the mother. Therefore, in the tarpana that comes as a substitute, the father is primary. The mother is with the father. In that regard, there will be no separate tarpana for the mother.
Even when doing shraddha for the mother, the homam is for the father. If the father is alive, there are rules that you should start with the grandfather and so on. Please ask the house pundit about all these.