Depending on the assist, you can combo into it a variety of ways. Each character tends to have a normal or spot between normals where you will find it comfortable to call an assist.
With Parasoul, you will see people call assists between the two hits of sHP sometimes, or after the second hit. There are other ways to call it though.
You can't call an assist DURING a special, but you can time an assist call WITH one.
Parasoul can do this by doing the Napalm Shot input alongside a button that would make it an assist call.
Something like charge back, forward+LPMK would do Light Napalm Shot + Assist.
This works with her other specials too, like tear toss or napalm pillar.
With Big Band, you can call it between the hits of cMP, or perhaps use your Sound Stun and call it after hitting the opponent with cHP.
Maybe, if you're in the corner, you could do sHK and call your assist, so that it hits them from the wallbounce sHK causes.
You'll find many creative ways to call your assists if you play around in training mode a bit. Just pay attention to the combo counter to make sure it actually did combo.
Also, make sure your training dummy is set to ground tech, so that you avoid any untrue combos!
You can turn on the hitstun bar too, that will help you see how much move a hitstun has.
Advanced training data will show it in terms of a number at the bottom of the screen.
And yes, you can combo into a tag! With Parasoul, try sHK in the corner into Big Band's tag, that might work.
Her backthrow probably leads into Big Band's tag too.
Big Band can likely do cHP with sound stun and tag to Parasoul to bring her in safely by comboing into it.