What can I call 2nd and 3rd place finishes in a competition?
A concise way to put it would be placegetter or placed. In the UK, Australia and New Zealand, "placed" would be understood to be in the top three. My understanding is a place in the US means first or second.
Medallist/medalled (UK spelling) or medalist/medaled (US spelling) might work if a medal was awarded.
One more possibility is podium finish - the first three in a motor sport event or cycling.
Even if metal discs weren't actually handed out, I'd have thought people would understand if you said silver and bronze.