Different learning traditions will offer up different advice. In the US, I learned that you only need to learn 4 principle parts, not 5. In particular, the second singular isn't necessary. The other four will suffice.
For third, third-io (I think what you call "mixed"), and fourth conjugations, the 1st person present and infinitive together clarifies which is which:
For third, it's curro currĕre
For third-io/mixed, it's facio facĕre
For fourth, it's audio audīre
If you add the second person singular present indicative active forms as well, the difference between the third/third-io and fourth is made clear by the long -i-:
3rd: curris
3rd-io: facis
4th: audīs
However, that's technically redundant, because you already get the differentiating element between 3rd-io and 4th in the infinitive (short e v. long i), and the differentiating element between 3rd and 3rd-io in the present (-o v. -io).