Can we do BEd regular course and Part time Ph.D. together

As per UGC stipulation in the matter one is not entitled to for two regular courses simultaneously. To do one regular course and one correspondence or distance education course is allowed under it. So you have to see that your PhD pursuit should not interfere with your B.Ed. in any case. You have to fully devote for your full time B.Ed. and then you can do PhD on the sidelines of that. One thing which you have to find out from the university from where you are pursuing PhD whether it is a part time distance course or a regular course because after completion of your PhD they will award you the PhD degree in that fashion only and it should not be mentioned in that as regular one as the documents should reflect it in the correct manner and should not violate the spirit of UGC in this matter. Please be alert and vigilant on that account not to acquire multiple degrees in manupulative ways.

Part-time PhD will be allowed only to employed people. They will not allow students to do a PhD on a part-time basis. Part-time PhD is not a distant course. As far as my knowledge goes in India no university is giving Ph D on distance mode. People who are working as a lecturer or assistant professor in any college or university will be allowed to do a PhD. They have to attend university for doing their work in a year for a minimum number of days.

You can't do a PhD while doing your B.Ed on a regular basis. You can do Ph D in regular mode and you can do your B.Ed on distance mode. There will not be any objection from the UGC. But what is the eligibility criteria for B Ed on distance mode is to be confirmed.

Travelling on two boats at a time is never advisable. You can't complete your Ph D and you can't concentrate on B.Ed. So select one course and do it.

Yes, you can take up both part time Ph.D and your B.Ed from the distance mode.
Part time Ph.D does not signify that it is from the distance mode but to be precise, you can enjoy this status because of your regular employment oppurtunity in the engineering college. Basically the persuasion of teaching in the engineering - college and persual of Ph.D simulteneously would consume a lot of time and even such a stage may develop where you will be frustrated due to time constraint.
This is one part and the other part would be to spare some time for the persual of research activity on the regular basis for some perod in a year prescribed by the university norm.
The existing situation would not allow you to undertake regular course of B.Ed .
Of course, undertaking the course from the distance mode like IGNOU seem to be a practical solution.
However, your involvement in that way would affect all your assignments and hence chalk out a plan prudently. In my opinion, taking up part time Ph.D and your present teaching assignment can be managed well. Later, you can appear for NET so as to get a chance of lecturer in a constituent college.

As per the rules and regulations laid down by he University Grants commission (UGC) one cannot attend two regular courses at the same time. Accordingly you have to select these courses in that combination only. PhD is a long process and B.Ed. can be done in only 2 years. So what I would suggest is first complete the B.Ed. and then you can register for PhD and meanwhile if you get a teaching job you can very well join that. This would be a better option for you for making a career suitably.