Yes, Hindu Woman can light funeral pyre of her father or
mother. Because it is clearly mentioned in the Vedas, the holiest scripture of
Sanatana Dharma (Hinduism or Hindu Religion), that there should be no
discrimination – not only among human beings but among all living beings.
First mantra of the Isha Upanishad of Shukla Yajur Veda extols the equality of
not just all human beings but of all living beings.
Ishavasyam Idam Sarvam Yat Kim Cha Jagatyam Jagat
Tena Tyaktena bhunjitha, ma gradha kasyasvid dhanam
Know that all this whatever moves in this moving world is enveloped by God.
Therefore find your enjoyment in renunciation, do not covet what belongs to others.
Gandhiji Explained this first mantra of Isha Upanishad like this:
Since God pervades everything, nothing belongs to you, not even your own body. God is the undisputed unchallengeable Master of everything you possess. If it is universal brotherhood – not only brotherhood of all human beings but of all living things – I find in this first Mantra of Isha Upanishad.
Since he pervades every fiber of my being and of all of you, I derive from it the doctrine of equality of all creatures on earth.
Garuda Puran is usually quoted to prove the point that only male members can perform last rites. But it must be noted that Garuda Puran does not state that women should not perform the last rites. Male members performing the last rites are a social custom which is directly related to property rights. It has nothing to do with Hindu religion.
Sons or daughters they all are the same in Hinduism. There
is no difference – what a son can do that a daughter too can do. All that you
need is the courage to stand up and protect dharma - the equality of all living
beings.
Hinduism is a living religion. It is constantly updated and
reformed. It is not stuck to any one scripture. There is permanent reformation
in Hinduism. If anything is found which conflicts with facts, it is quietly
forgotten.