That is why we are learning chess openings: If you know the opening, you will be able to quickly answer with a best move without loosing your play time. If you don't know the opening, you have to find it at the chess board... You may find a best answer, you may not...
1. The stated line 1.e4 e5 2.Nf3 Nf6 3.Nxe5 Nxe4 4.Qe2 d5 5.c4 Bc5 6.cxd5 Bxf2 may show that neither White nor Black has no essential initial knowledge on correctly playing the Petroff Defense.
2. Within the line, Black was first to make the mistake:
After 1.e4 e5 2.Nf3 Nf6 3.Nxe5,
Black must never play 3... Nxe4 because White will be better in all the cases. On this, you are recommended to see this short video (find it at the bottom of the page open by following this link):
Petroff Defense Trap
3. Then,
White made the mistake by moving 5.c4. Instead of 5.c4, White should play 5.d3 Qe7 6.dxe4 Qxe5 7.exd5 with one extra pawn and better position, at least.
4. After 1.e4 e5 2.Nf3 Nf6 3.Nxe5 Nxe4 4.Qe2 d5 5.c4 Bc5,
instead of 6.cxd5 White should play 6.d3 Bxf2+ 7.Kd1 Bc5 8.dxe4 dxe4+ 9.Kc2 Qe7 10.Bf4 Bd6 11.Nc3 Bxe5 12.Nd5 Qe6 13.Qxe4 Bxf4 14. Qxf4 with equal position.
5 After 1.e4 e5 2.Nf3 Nf6 3.Nxe5 Nxe4 4.Qe2 d5 5.c4 Bc5 6.cxd5 Bxf2, the position may be estimated as "Black is much better" (not just "Black is better"). And the only thing for White is to go on with 7.Kd1 Qxd5 8.Qb5+ Qxb5 9.Bxb5+ c6 10.Bc4 and to hope for any possible mistake from Black at a later stage...