Rarity is printed on the card and has no effect on gameplay(Comprehensive Rules 2.3.8). Parallel versions have the same card number as the base card, so they count toward the same 4-copy deck limit (Comprehensive Rules 4.1.3).
The ladder, lowest to highest
C: Common
U: Uncommon
R: Rare ← this card
RR: Double Rare
RRR: Triple Rare
RRRR: Quadruple Rare
ExP: Parallel ← this card
AP: Parallel
SP: Parallel
SSSP: Parallel
UR: Parallel (highest)
This order comes from the official card database’s rarity list. The base tiers are C, U, R, RR, RRR and RRRR; ExP, AP, SP, SSSP and UR are parallel versions. Official full names are given only for the base tiers. SEC shows up on at least one card (BP08-SEC) but isn’t in the official database list, and we couldn’t verify an “EXC” rarity, so neither is placed on the ladder.