How Do I Improve the Hourglass Shape of my Face? (photo)

Young patients with fullness of the lower anterior cheeks (chubby cheeks) usually have enlarged buccal fat pads. If the middle of the face is thin, it contributes to an "hour glass" facial appearance. 
Chubby cheeks usually run in the families ( it is inherited). Chubby cheeks usually remain present along life and tend to worsen with aging (e.g. patients look chubbier compared with young pictures of them).
Buccal fat removal is the plastic surgery procedure that permanently removes the fullness of the lower anterior cheeks (chubby cheeks). It allows patients candidates to achieve a thinner, V-shaped, chiseled facial appearance like models. I do the procedure under local anesthesia without pain under 45 min in the office setting ( no need to go to sleep). Results start being seen in 3 weeks. Final results show in 3 months post procedure. 
The video below helps to understand who could be a candidate for buccal fat removal. 
John Mesa, MDBuccal Fat Removal Specialist Plastic Surgeon New York City - New Jersey - Miami

There are some modest improvements you can make to your facial shape but jawline reduction/shaving is not one of them. You may consider partial buccal fat removal and perioral liposuction to reduce the convexity between the cheek bones and the jawline.

Jawline/Cheek reduction can be complex so be careful. There are 4 anatomic layers that can contribute to a full cheek or "hourglass" wide lower face: the Fat, the Masseter muscle, the Parotid salivary gland, and the Jawbone. Options exist to reduce each of these layers, but often it is better to augment your cheekbones higher in your face in order to restore a more balanced look. My advice would be to seek out a good plastic surgeon in your area who you can spend an hour with you to discuss these issues.  

 Aesthetically speaking the photos show flat cheeks, excess subcutaneous (I call it baby fat) and a weak chin.  Removing the buccal fat is not recommended as it can cause facial indentations making the face appear gaunt when you are in you 30's.  You can have Cheek Augmentation with micro-liposuction to remove the subcutaneous fat in the face along with Chin Augmentation using a Chin Implant.  These would create a naturally more attractive and feminine face IMHO.  Hope this helps and please share this answer with your parents.