As well as general traineeships, there is also a scheme of translation traineeships for university graduates.
Here is an extract from the Parliament website:
These traineeships are awarded only to graduates of universities or equivalent institutions. Their purpose is to enable trainees to supplement the knowledge acquired during their studies and to familiarise themselves with the activities of the European Union and, in particular, the European Parliament.
Applicants for a translation traineeship for university graduates must:
be nationals of a Member State of the European Union or a candidate country;
be aged 18 or over on the traineeship start date;
have obtained, before the deadline for submission of applications, a university degree following a course of study of at least three years’ duration;
have a perfect knowledge of one of the official languages of the European Union or the official language of an candidate country and a thorough knowledge of two other official languages of the European Union;
not have been awarded any other traineeship or have been in paid employment for more than four consecutive weeks at the expense of the European Union budget.
Translation traineeships for university graduates are awarded for a period of three months. Exceptionally, they may be extended for a maximum period of three months.
The next application period goes from 15 June to 15 August (midnight), for traineeships starting 1 January, with a note to this effect: “We advise you not to wait until the last day to apply, as a large number of applications is liable to overload the system.”.
Find out more here, scrolling down to “Translation traineeships”, or indeed also “Translation training placements”, or for general training “Traineeships for university graduates – General option”.