Last time I looked at Mailjet they didn't offer this. Sendgrid does do that, as does Mailgun. I've tried both of them for BunkerApp and decided on Mailgun for various reasions. One reason was that Mailgun offered a simple way for a catch-all-receive (* @*.bunkerapp.com) while with Sendgrid I had to jump through houps. Mailgun also offers a (for my needs) simpler parsing API.

If you need Inbound mails I'd definitely recommend looking at both Sendgrid and Mailgun (and Mailjet if they offer this now), then decide on what suits your particular needs more.

I asked a couple of weeks back and they did not, at least yet.

I'm not sure, as I never needed to do that.

no email parsing. we use a free mailgun plan for that