Is it permissible to pray with a mother who has suffered a stroke and partial paralysis, after she has performed tayammum and greeted us?

Your mother must pray according to her ability as long as she is conscious. She should perform ablution if possible, otherwise, perform tayammum (dry ablution). The recitation of Al-Fatiha is waived for her if the Imam recites it. As for the remaining remembrances, she should say them in her heart if she is unable to utter them silently.