This can be an issue. It isn’t always the case, and it will depend on the firm in question and also who you are working for as a paralegal. The more the firms are direct competitors, then the more this can be an issue.
I have seen enough paralegals do vacation schemes for it to be okay though. But have seen enough paralegals not get approval from either their current employer or the vacation scheme firm due to their “you can’t work for another employer at the same time” policies or because a conflict check has made it too risky.
The firm won’t do a conflict check until you start an onboarding process though, and so you can be progressed and then find out you cannot proceed. You’d also need to declare it to the firm you are paralegalling with too, which means they could also block it.