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First of all karan never claimed that he expects tulsi to love him like her own children , rather it's tulsi herself who says i have never differentiateted between karan and my own children.
Secondly karan clearly told nandini that rio and niyati r his responsibility to him. He raised rio so there was an attachment , but he never said he loves rio the same way he loves parth and samaira . Even when tulsi wanted to get vaishnavi and rio married , he told tulsi his decision was wrong and just sided with parth. Even when karan mentioned taking parth and nandini back to the usa as soon as they return from khandala, niyati said what will happrn to rio and me ? Karan said tulsi maa will take care of you two. So he feels a much deeper grief over parth's death.
Lastly in season 1 karan loved both bhoomi and manthan but he loved bhoomi more. Post leap when he used to live like a devdas for nandini karan told tanya that bhoomi was his only reason to live , a sign of nandini.
What can I say, if he always loved Bhoomi more than Manthan, then he was a bad father that season too.
Every parent can have a favorite child, but preferring a child over other is the worth sign of being a parent.
About this season, Rio had always looked after him as father and he raised him, if he differentiated between them.
Aside he did say that Tulsi is being biased against Parth
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