, then your are not directly liable for violating any employment contract terms.
However, the Law of Country mandates an application to be processed from you with the signature of your current sponsor and new employer which requires your part-time service agreeing you to work for the other. This need to be approved by Govt. authorities, thence you can work for that other party.
In case, by Part-time job what you meant was "taking any home tuition" .... it seems less chance of getting into troubles and issues !!
Good luck !! & God bless you !!