In theory there's a directorate of the EU parliament called Directorate-General for Security and Safety (or DG SAFE for short) whose description is:
The Directorate-General for Security (DG SAFE) works to facilitate
Parliament activities, while guaranteeing sufficient protection to
people, assets and information.
And it's main tasks are:
- providing the most appropriate security and safety services to Members and the institution;
- deploying appropriate security measures, based on applicable security policies and sound risk assessment;
- facilitating accreditation and access to Parliament premises and welcoming a wide range of occupants and visitors, including - Members
and their entourage, Parliament and other EU institutions officials,
service providers, journalists, interest group representatives,
diplomats and general visitors;
- deploying and managing innovative and efficient physical access control systems;
- communicating, training and informing on security and safety matters;
- cooperating with all entities and groups to enhance security posture and create a positive security culture;
- assuring security, safety and prevention needs 24 hours a day, ensuring remote
- monitoring of buildings and technical installations.
From all the EU directorates this should be the one dealing with this subject. However it likely does not exist a security network like the NATO one mentioned by MEP Sandra Kalniete:
“Every NATO employee or intern, as they start work at the NATO
Headquarters, is required to receive a number of security clearances
from the relevant agencies. Unfortunately, the European Parliament has
no such security net. Until now, we have been ignoring security
considerations, leaving it to each individual MEP to pick the people
they want to hire as employees or interns at their offices. However,
the internship of Putin press secretary Peskov’s daughter is contrary
to any security standards,” Kalniete said.
The problem, however, is not clear cut. Elizaveta Peskova is not an accredited assistant, but a trainee.

Trainees are people who have signed a traineeship agreement with a
Member. Traineeships contribute to education and European vocational
training and promote a better understanding of the workings of
Parliament. They can take place either on Parliament’s premises or in
the Member State in which the Member was elected.
As so it is unlikely that Elizaveta Peskova on her own has any kind of access to relevant information. That is not the problem. The problem is that the person that put her as trainee is Aymeric Chauprade, a politician known for his connections to far-right movements, including France political party National Front. He has access to those informations and he has supported Russia in the past including the annexation of Crimeia:
“Russia has become the hope of the world against new totalitarianism,”
Mr. Chauprade, the National Front’s top European Parliament candidate
for the Paris region, said in a speech to Russia’s Parliament in
Moscow last year.
When Crimea held a referendum in March on whether the peninsula should
secede from Ukraine and join Russia, Mr. Chauprade joined a team of
election monitors organized by a pro-Russian outfit in Belgium, the
Eurasian Observatory for Elections and Democracy. The team, which
pronounced the referendum free and fair, also included members of
Austria’s far-right Freedom Party; a Flemish nationalist group in
Belgium; and the Jobbik politician in Hungary accused of spying for
Russia.
So, unfortunately, the security liability exists with or without Elizaveta Peskova.