I am paying medical insurance premium for a medical policy taken in my name, my wife and children and also paying premium on a medical policy taken in the name of my parents who are above 70 years. Can I claim a deduction for both premiums paid?

As per Section 80D: –

A deduction to the extent of Rs. 25,000 is allowed in respect of premium paid to effect or to keep in force an insurance on the health of self, spouse and dependent children.

A further deduction up to Rs. 50,000 is allowable to effect or to keep in force insurance on the health of Senior Citizen Parents (above 60 years) of the assessee.

Hence the assesse can claim a deduction for both premium paid.