I'm slightly wondering how can Amazon send me a prime parcel, where the goods is 5 EUR.

Lots of replies but I see that no one has answered your question directly. Here in the US, Amazon maintains warehouses all over the country and they use their own vehicles or a sub-contractor (usually a free lance individual) to deliver their items from the nearest warehouse. They don't use the US PS, UPS, FexEx or any of the other usual shipping companies so that saves them tons of money.
And BTW the price that most companies pay for shipping via those companies is a small FRACTION of what the same shipping company charges individuals!
I'm sure it does. Here it is usually UPS, sometimes PostNL delivering the packages. Judging by the fact that Amazon FBA is about 4 EUR for such small items, pf course they get the preferential treatment.
You do realize that a $1 parcel from China does not run the shortest path, right? They are likely to be routed through south east Asia or HK or who knows where, stay there and wait to catch the cheapest under loaded flight and that's why they arrive in months, even by air.
If I am to ship a small board to Europe or NA with a reliable, predetermined schedule, I pay MORE than your 13 Euros. DHL charges around $25 per package, and SF-Express charges $15 per envelope, $25 per package, and China Post EMS charges $30+ per package, both quoted for 1kg rest of the envelope.
They often use Cainiao Super Economy, which is Alibaba's own shipping service. I'm sure a lot of it is in 30ft containers on a ship, that's why they have 30-50 days delivery times.
Its the same here in Canada, sending a letter is cheap, but any sort of small package is ~$15. So small sales of t-shirts, PCBAs, etc. is just not realistic.
You can look if there are any companies offering discounted shipping rates. The one I use here is Netparcel.
When I was a student at university, I regularly traded online. I was trading usually phone accessories, buying bulk on ebay and selling it on the local ebay equivalent. It was a nice income, paying more than the scholarship. Everything was in padded envelopes, and the only times I had issues with the shipping charges, when someone asked for collect-on-delivery. But they were paying for that, so no worries.
Here is what really pisses me off: Sending a letter, 20-50 grams is 3.1 EUR, worldwide.
If I place a PCB in the said letter, it starts at 13 EUR.