Amish mentions I cannot in 121 but I am operating under Part 135, and this also contradicts Eagle06's post. Thank you
Yes, you can log it. No you should not.
Under part 61, you can log the time, regardless of whether the operating rules for your flight are Part 91, Part 121, or Part 135. However, you really should't, if you're operating for a certificate holder, for a three reasons.
When you're logging time as a professional, working for a certificate holder, you're going to be using that time at some point in the future to represent yourself to a prospective employer. The prospective employer wants to know about your real experience, not what you could legally get away with logging. The new employer will have the ability to question your old employer, even see your records, and will know how much flight time the old employer says you have. If you claim 1000 hours of PIC time, but your old employer shows 500 hours of PIC and 500 hours of SIC, then it doesn't look good for you; you appear to be inflating your logbook.
In the 121 and 135 world, PIC means time when you're really the PIC. This isn't regulatory; this is an understanding industry-wide. While the regulation does indeed say that there will only be one PIC and that's the one assigned by the employer, and that the PIC remains the PIC for the duration of the flight...Part 61.51 still provides that you can log it as PIC...but the industry at large doesn't see it this way.
If you're logging time as PIC when you're not the person who signed for the airplane, and who was the designated PIC for the operation, it reflects badly on you. You will appear to be inflating your times or padding your logs. You will also appear to not understand the relationship between PIC/SIC and 121/135...which suggests a certain level of ignorance to a prospective employer...at a minimum it leaves a bad impression. Finally, it may cost you a job, or when the prospective employer learns of this perceived deception, the times you have listed will be devalued or reduced.
Having said that, many employers use varying means to record times in their applications...what one counts, another doesn't. Some will consider military time plus an additional factor, for example, others won't. Some will lump all time in piston airplanes into one category, even if it's a large, four engine airplane. One thing that's universal, however, is that if you haven't signed for the airplane (if you're not the legal PIC), the employer doesn't count the time as PIC. If you've logged it as such, it will reflect badly on you.
The chief purpose for the rules allowing the logging of time the way they do under Part 61.51 is to permit recognition for various types of flying while pilots work on their certificates or ratings. While it's perfectly legal to log the time as sole manipulator while employed as SIC under Part 135, it's viewed as an inappropriate act and exploiting a "loophole." It's best to bide your time, log SIC when you're SIC, and the when you upgrade, log PIC.