As far as I'm aware, at least some schools of law (in particular, the Shafiis) say that qisas does not apply in the case of a parent killing his/her child, and only diyya applies. I can't seem to find any sources outlining who is entitled to receive diyya in this case? I don't know how to determine the recipient of diyya/the people having the right to waive qisas in murder cases in general, but this particular constellation is probably different from other constellations.
A parent is not killed in Qisas (retaliation) for their offspring according to the Shafi, Hanafi and Hanbali schools of thought.
Instead Diyya (blood-money) is taken from the killer and given to the remaining heirs of killed person.
رجل من بني مدلج قتل ابنه فأخذ منه عمر مائة من الإبل ثلاثين حقة وثلاثين جذعة وأربعين خلفة فقال ابن أخي المقتول سمعت رسول الله صلى الله عليه وسلم يقول ليس لقاتل ميراث
A man from Banu Mudlij, killed his son, and 'Umar took one hundred camels from him, thirty Hiqqah, thirty Jadha'ah and forty Khalifah. Then he said: “Where is the brother of slain? I heard the Messenger of Allah (ﷺ) say: 'The killer does not inherit.'"
ويجب على الأب الدية للورثة، ويحرم منها
Blood money is obligatory on the father, will be given to the heirs, and is forbidden for him (the father).
According to the Maliki school of thought Qisas applies in this situation if it was intentional murder.