In a certain code language, "SPARROW" is written as "1326654", and "RING" is written as "6978". How is "RAINS" written in that code language?

This is a direct letter substitution cipher. Let's map the digits to letters from the given words:
From SPARROW → S:1, P:3, A:2, R:6, O:5, W:4
From RING → R:6, I:9, N:7, G:8

Now find the code for RAINS:

R → 6

A → 2

I → 9

N → 7

S → 1

Putting them together, RAINS is coded as 62971.