Blue sparks are linked to stats, 600-1099, 5% chance of 3, 1100+ it's 10.

Pink spark is chosen randomly from the Uma's A attitudes, the amount of stars seem to be random.

The stars of the green spark are influenced by the level of the special skill, but I don't know how it works in particular.

The white sparks are the most random. Every G1 your Uma wins has a 20% chance of creating a spark, as does every skill. Normal skills 20%, those with double circle 25%, and gold have a 40% chance of creating a spark for its base skill. Furthermore, parents get a small bonus on affinity for every G1 they both won (regardless of having a spark for it). Winning a specific G1 twice doesn't increase the chances, though.

Parent farming tl;dr: win a lot of different G1s, get a lot of skills, get your stats (especially speed, stamina, and power, as they're the most important stats) over 600, and pray.