I want to travel to the Schengen area in 13 days, but it takes 15 days to issue a visa. Can I still apply?

My parents are visiting Europe for 2 weeks (more details below). I currently live in the US but I'm hoping to join them on this trip. My parents made some changes to their itinerary due to availability constraints which put me in the following situation (as I already had scheduled my visa appointment):

The Norway visa processing time according to their website is 15 business days. My intended travel date is on the 13th business day which gives me only 11/12 business days for the visa processing. No earlier appoinments are available. I still want to go ahead and take chances. My questions

1) How would they deem such travel plan if it starts before claimed visa processing time.

• Are they going to reject it?
• Are going to grant visa for a different start date according to their processing completion but stick to the end date according to travel plan?

2) Here's a short version of itinerary: first 5 days in Norway, next 2 1/2 days in Germany, the rest is 1-2 days in a few other countries. I'd still love to join my parents even when their trip is 25% through. How do I convey that in my application.

I'd really appreciate your comments.

Update

Happy to share that I got the visa on 7th day (not 7th business day, just 7th day). I applied for Norway visa in San Francisco VFS office.