ECC is indeed used by CloudFlare's website but only for the session key agreement. The authentication is performed using an RSA 2048 bit private key. The corresponding RSA public key is in the certificate. In other words, although ECC is being used, it is not used for authentication and therefore not part of the certificate.
The ciphersuite is:
Which means TLS with ECDHE (ephemeral Elliptic Curve Diffie-Hellman) key agreement, RSA based authentication (and thus certificates). The ephemeral part means that no static private ECDH key is used on the server and no public ECDH key is therefore present in the certificate.
Or, as specified by RFC 4492 section 2.4:
This key exchange algorithm is the same as ECDHE_ECDSA except that
the server's certificate MUST contain an RSA public key authorized
for signing, and that the signature in the ServerKeyExchange message
must be computed with the corresponding RSA private key. The server
certificate MUST be signed with RSA.
To state that the CloudFlare website is using ECC certificates (not certification, that term means something else) is therefore incorrect. It uses a cipher suite that performs ECDH key agreement.
The used explicit or named parameters are determined during the handshake. The client has the ability to include a list of supported curves, ordered to preference RFC 4492 section 5.1.1. The point format - uncompressed (default) or compressed - can also be indicated. As the EC points themselves are ephemeral (short-lived) you can only retrieve them by analyzing the handshake.
For completeness the rest of the ciphersuite: AES-128 bit encryption using GCM authenticated (AEAD) mode encryption. The final SHA-256 is used as definition of the keyed hash (HMAC) used for key generation (or key derivation) and validation.