In my opinion, you might be better off re-filing and setting the withholding allowances to 0 at the part time job. Almost surely, each firm's payroll software computes the taxes to without each pay period as if that job is your only job. If you are only working the second position part-time, you probably won't be getting very large paychecks from it. And, from that job's payroll software's point-of-view, you won't be earning much money, putting you in the lowest tax bracket. As it is the lowest tax bracket, the software will think that you won't have to pay many, if any, taxes to the government.
But we know that isn't true, because you have the other full time job. The government only cares about total income and paying your taxes on the total, not what each firm's software computes.
In fact, depending on how much will be withheld from each, you may even need to declare them both 0, or even ask for additional money to be withheld from each pay period. Furthermore, the onus is on you to make sure that you are paying enough taxes in a timely manner. If you complete 2017's tax forms next year and you underpaid by a significant amount, you may be subject to interest and penalties. It is unlikely in your situation that you will be far enough off to invoke those penalties, but you don't have to risk it, either.
I'd recommend that you refile the 2nd job to 0 allowances immediately, and then after a few pay periods, use one of several online tax calculators to estimate how much tax you are likely to have to pay in for 2017 and how the withholdings are stacking up. And if you are likely to come up short, adjust the first job to 0 or adjust one or both of them to withhold a little extra money per pay period. Another option, if you are going to come up short, would be to make proactive estimated tax payments.