Cfrgtkky

Austrian foreign minister Karin Kneissl was interviewed in Newsnight recently. When asked whether she could support a hypothetical request from the UK to extend Article 50, she did not immediately say no, but pointed at the European Parliament election which are held in May 2019, and said that it could therefore be extended by at most a couple of months anyway. Why would the European Parliament election be an objection to extending Article 50 beyond this date? Could not the United Kingdom participate in the elections, and keep its seats until the date of the extended Article 50, if it is indeed extended?

It's not a problem of rejection by the EU parliament but of legality. If an extension were to occur the UK, as an EU member, would have to have seats in the EU parliament. But when the UK cabinet made clear this would not happen the European Council Decision (EU) 2018/937 on establishing the composition of the European Parliament was made leaving the UK out.

You can find the table for 2019-2024 EU representatives in the EU parliament in Article 3:

Article 3

1. 
   

   The number of representatives in the European Parliament elected in each Member State is hereby set as follows for the 2019-2024
   parliamentary term:

   
   
   

   (...)

   
   

Yet a point to the same article was added in case the UK still remained a member by this point:

2. However, in the event that the United Kingdom is still a Member State of the Union at the beginning of the 2019-2024 parliamentary
   term, the number of representatives in the European Parliament per
   Member State taking up office shall be the one provided for in Article
   3 of the European Council Decision 2013/312/EU (4) until the
   withdrawal of the United Kingdom from the Union becomes legally
   effective.

But obviously we are getting closer to the EU elections so many parties in most members have already started preparing their campaigns counting on the new table. As so in case the UK still remains a member of the EU by 2019 the following applies (still Article 3):

All representatives in the European Parliament who fill the additional
seats resulting from the difference between the number of seats
allocated in the first and second subparagraphs shall take up their
seats in the European Parliament at the same time.

Which seems like a solution, even is somewhat bizarre, to the problem. The sticking point however is schedule. You see, in Article 4, it is stated that any modification must be submitted sufficiently far in advance prior to the election.

Article 4

Sufficiently far in advance of the beginning of the 2024-2029
parliamentary term, the European Parliament shall submit to the
European Council, in accordance with Article 14(2) TEU, a proposal for
an updated allocation of seats in the European Parliament.

Unfortunately no definition exists for what constitutes as sufficiently far in advance submission. As so the EU parliament and European Council would need to consider it. Strictly speaking there's no way for the Austrian Foreign Minister to know if this feasible or not by EU law alone.

The following article was quite useful in the writing of this answer:

Could Article 50 be extended to allow for a second Brexit referendum?

(...)

According to the EU’s arrangements, the reallocation of seats will not
take place if the UK remains a member. This would cause inconvenience
for those member states due to receive additional seats, which might
require arrangements to be put in place to fill them should Brexit
occur later. The sudden loss of 46 seats and reallocation of 27 others
sometime after the elections, almost certainly resulting in a degree
of political rebalancing, could potentially have some destabilising
effects in the European Parliament. A second worry on the EU side is
that UK participation in the MEP elections in controversial
circumstances could result in election of a large number of
Eurosceptic members, again with potentially destabilising effects,
even if their presence is only temporary.

(...)

The European Parliament is made up of elected representatives from member states. If the UK was still in the EU at that time, it would have to run the election and send the winners to the EU Parliament. They would get a salary and pension rights, staff and more.

This burden on both the UK and EU would waste a lot of money and require things like the so-called divorce bill to be recalculated.