Piston Rod Bearing?

I'm thinking the mechanic you chose hasn't a clue. Lifter noise, especially at start up, is quite normal. In fact my car has had that tick noise since the day i drove it off the lot. It's even more prominent now that it's getting cold. It does go away however once warmed up as it should.

Rod bearing's, when failing, is more of a thunk than a tick. Without hearing it in person however it's nearly impossible to even guess as to what the sound is your hearing.

Have you checked your oil level? Any way you can get the sound recorded and post here?

 

How many miles on the car?

The 2013 C uses solid lifters not hydraulic lifters. Solid lifters are very noisey like dubit said especially when cold. And like dubit said a rod bearing is a knock not a tick. Your mechanic is pulling your you know what because in order to replace the solid lifters both cams have to come out pretty big job & expensive @ $80 an hour. Bet he didn't do anything to it.

Go to harbor freight and buy a mechanics stethoscope. Not sure how to put a 2013 in inspection mode like you can with a G2 where the engine will run continuously for inspection or get someone to keep the gas pedal depressed while you listen around in the engine with your scope. I think they deleted that Inspection mode on newer models.
If its ticking its probably coming from the valve cover area.

before you do anything else buy some really high quality oil and try it. This is the oil on I use on my car and it has quieted down the lifter tick significantly.

Red Line Synthetic Oil. 10W30 Motor Oil

Free Shipping on Orders Over $99 at Summit Racing

Read the reviews on the Summit site about this oil helping lifter tick. It has an excellent ZDDP package to to help with engine wear. Best synthetic oil on the market.

Buy the 1 gallon your car holds 3.9 quarts.

here's pictures of my G2 engine with 150,000 miles on it using Redline oil. Same engine that's in your car. A you can see all that has to come out to get to the solid lifters under the cam journals.
Also note at 150,000 miles zero wear on the cam journals the bright shiny round things that push down on the lifters.
Its unlikely the cam/lifter buckets are worn unless the car has very high mileage or seen horrible oil maintenance like free dino oil changes at the dealer and then hardly ever changed.
People are notoriously bad about maintaining a prius engine based on what i see on this forum.

Go on toyota.com/owners forum and join. Have your insurance card handy as it will require the cars vin.
Once properly joined you will be able to see every event the car has had at an authorized Toyota dealer. Many times the dealer reported bad news and the car gets sold. You will see any bad news on that service forum.

 

That some excessive valve noise the first one in the cabin that's loud. You should not hear that with engine rpm so low. You should only hear it if you get on it pretty good. You may have to take the valve cover off and take a look in there. Again.... the mechanic replaced nothing.

Well if you throw Midas the keys and say fill her up they will use some cheap dino oil. It will sound like an industrial sewing machine. The solid lifters in our car are loud when cold and going through its different stages of warm up. Even louder with 20 weight oil. In my opinion 20 weight oil is way to thin for a solid lifter car.
My opinion is correct I have pictures to prove it.

Buy that oil I showed you in 5-30 or 10-30 weight. Its essentially a 40 weight oil when warm. It will be night and day engine difference in both sound and performance.

Go to Jiffy oil change go online for coupon bring them the gallon jug of oil I showed you and a new filter. Around here its $24. labor. That's it.
They will say hey it takes 5 quarts cause there database is wrong just say like I do that's nice but today we are using 4 quarts.

Good Luck!

 

I'm going to go the opposite way of the reply above. While I don't think a heavier oil will hurt, I have to be honest and tell you that in your video - the noise is no different than what I hear myself.

Video 1 - That's right after startup, throwing it into drive and going. Correct?
Video 2 - Cold engine. May have been driven some that day, but has sit long enough to make the noise.

I'd say change the oil if you like. But I don't think anything is wrong here.

 

Agreed. Very loud. Something bad there.
That’s why I said remove the valve cover but the op probably won’t know what to look for other than horrible sludge.

I should have said After you fix this handle the oil change like this.