score:3
I changed their flight to Houston from Istanbul to next week ($250 fee) so that she could get a surgery in Istanbul and I could think about what to do with the flight ticket.
Your first step should be to contact your parents' travel insurer, who can probably re-arrange your trip for when they are both back on your feet, and directly arrange refunds and compensation for them.
If they are not helpful, then in terms of refunding a partially flown ticket, the cheapest TK fares on Karachi-Houston include the following provision:
CANCELLATIONS
TICKET IS NON-REFUNDABLE IN CASE OF NO-SHOW.
NOTE - TEXT BELOW NOT VALIDATED FOR AUTOPRICING.
IN THE EVENT OF NO SHOW - WHEN REFUND IS
REQUESTED AFTER DEPARTURE OF THE ORIGINALLY
SCHEDULED FLIGHT-
NOT PERMITTED EVEN FOR MEDICAL REASON
FOR NON REFUNDABLE TICKETS THE -YR- SURCHARGE
WILL NOT BE REFUNDED.
What this means in particular is that you must cancel your ticket and apply for a refund before your scheduled flight is due to depart. (That means your newly changed flight from Istanbul to Houston.) If you simply "no show" (fail to turn up without telling the airline), then your ticket will become valueless.
Continuing to read the penalties category,
AFTER DEPARTURE
CANCELLATIONS PERMITTED FOR CANCEL/REFUND.
NOTE - TEXT BELOW NOT VALIDATED FOR AUTOPRICING.
FOR CANCELLATIONS AFTER DEPARTURE THE REFUND GIVEN
WILL BE THE DIFFERENCE BETWEEN THE FARE PAID AND
THE APPLICAPLE FARE FOR THE JOURNEY FLOWN.
APPLICABLE FARE FOR THE JOURNEY FLOWN IS THE ONE
THAT SET AT THE SAME RBD WITH THE ORIGINAL FARE
PAID.
IF NO FARE EXISTS FOR THE FLOWN JOURNEY
ON THE SAME RBD WITH THE ORGINALFARE
ANY FARE SET ON A HIGHER RBD WILL BE ACCEPTED.
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Provided you cancel in advance, you need to find the lowest possible one-way fare from Karachi to Istanbul in the same booking class (RBD) as your parents originally flew, calculated on the day they originally purchased the ticket. That is the cost of your one way flight; any remainder would be your refund. A service fee may also apply from your travel agent.